CHAPTER 1: Functions 1.1: Functions 1.2: Graphs of Functions 1.3: Basic Graphs and Symmetry 1.4: Transformations 1.5: Piecewise-Defined Functions; Limits and Continuity in Calculus 1.6: Combining Functions 1.7: Symmetry Revisited 1.8: x = f ( y) 1.9: Inverses of One-to-One Functions 1.10: Difference Quotients 1.11: Limits and Derivatives in Calculus Functions are the building blocks of precalculus. In this chapter, we will investigate the general theory of functions and their graphs. We will study particular categories of functions in Chapters 2, 3, 4, and even 9.
SECTION 1.1: FUNCTIONS (Section 1.1: Functions) 1.1.1 LEARNING OBJECTIVES Understand what relations and functions are. Recognize when a relation is also a function. Accurately use function notation and terminology. Know different ways to describe a function. Find the domains and/or ranges of some functions. Be able to evaluate functions. PART A: DISCUSSION WARNING 1: The word function has different meanings in mathematics and in common usage. Much of precalculus covers properties, graphs, and categorizations of functions. A relation relates inputs to outputs. A function is a relation that relates each input in its domain to exactly one output in its range. We will investigate the anatomy of functions (a name such as f, a function rule, a domain, and a range), look at examples of functions, find their domains and/or ranges, and evaluate them at an input by determining the resulting output.
(Section 1.1: Functions) 1.1.2 PART B: RELATIONS A relation is a set of ordered pairs of the form ( input, output), where the input is related to ( yields ) the output. WARNING 2: If a is related to b, then b may or may not be related to a. Example 1 (A Relation) Let the relation R = {( 1, 5), (,5), ( 5,7) }. ( 1, 5) R, so 1 is related to 5 by R. Likewise, is related to 5, and 5 is related to 7. R can be represented by the arrow diagram below. PART C: FUNCTIONS A relation is a function Each input is related to ( yields ) exactly one output. A function is typically denoted by a letter, most commonly f. Unless otherwise specified, we assume that f represents a function. The domain of a function f is the set of all inputs. It is the set of all first coordinates of the ordered pairs in f. We will denote this by Dom( f ), although this is not standard. The range of a function f is the set of all resulting outputs. It is the set of all second coordinates of the ordered pairs in f. We will denote this by Range( f ). We assume both sets are nonempty. (See Footnote 1 on terminology.)
(Section 1.1: Functions) 1.1.3 Example 2 (A Relation that is a Function; Revisiting Example 1) Again, let the relation R = {( 1, 5), (,5), ( 5,7) }. Determine whether or not the relation is also a function. If it is a function, find its domain and its range. Refer to the arrow diagram in Example 1. Each input is related to ( yields ) exactly one output. Therefore, this relation is a function. Its domain is the set of all inputs: { 1,,5}. Its range is the set of all outputs: { 5, 7}. Do not write { 5, 5, 7}. WARNING 3: Although a function cannot allow one input to yield multiple outputs, a function can allow multiple inputs (such as 1 and in Example 2) to yield the same output (5). However, such a function would not be one-to-one (see Section 1.9). Example 3 (A Relation that is Not a Function) Repeat Example 2 for the relation S, where S = {( 5, 1), ( 5, ),( 7,5) }. S can be represented by the arrow diagram below. An input (5) is related to ( yields ) two different outputs (1 and ). Therefore, this relation is not a function.
(Section 1.1: Functions) 1.1.4 TIP 1: Think of a function button on a basic calculator such as the x 2 or button, which represent squaring and square root functions, respectively. If a function is applied to the input 5, the calculator can never imply, The outputs are 1 and. Example 4 (Ages of People) For a relation R, The set of inputs is the set of all living people. The outputs are ages in years. ( a, b) R Person a is b years old. Is this relation a function? Yes, R is a function, because a living person has only one age in years. Example 5 (Colors on Paintings) For a relation S, The set of inputs is the set of all existing paintings. The outputs are colors. ( a, b) S Painting a has color b. Is this relation a function? No, S is not a function, because there are paintings with more than one color.
PART D: EVALUATING FUNCTIONS (THE BASICS) (Section 1.1: Functions) 1.1.5 If an input x Dom( f ), then its output is a well-defined (i.e., single) value, denoted by f ( x). We refer to x here as the argument of f. We refer to f x f x ( ) as the function value at x, or the image of x. ( ) is read as f of x or f at x. ( ) does not mean f times x. WARNING 4: f x A function can be modeled by an input-output machine such as: x f f ( x) When we evaluate a function at an input, we determine the resulting output and express it in simplified form. A function is defined (or it exists) only on its domain. If x Dom( f ), then f ( x) is undefined (or it does not exist). Example 6 (Evaluating a Function; Revisiting Examples 1 and 2) Let the function f = {( 1, 5), (,5), ( 5,7) }. a) Evaluate f ( 5). b) Evaluate f ( 6). a) ( 5, 7) f, so 5 Dom( f ), and f ( 5)= 7. b) 6 Dom( f ), so f ( 6) is undefined.
(Section 1.1: Functions) 1.1.6 Example 7 (Failure to Evaluate a Non-Function; Revisiting Example 3) Let the relation S = {( 5, 1), ( 5, ),( 7,5) }. If we had erroneously identified S as a function and renamed it f, we would see that f 5 ( ) is not well-defined. It is unclear whether its value should be 1 or. A function rule describes how a function assigns an output to an input. It is typically given by a defining formula such as f x ( )= x 2. Example 8 (Squaring Function: Evaluation) Let f ( x)= x 2 on. This means that we are defining a function f f ( x)= x 2, with Dom( f )=. by the rule The rule could have been given as, say, f ( u)= u 2. Either way, f squares its input. Evaluate f ( 3). We substitute 3 ( ) for x, and we square it. f ( x)= x 2 f ( 3)= ( 3) 2 = 9 3 f 9 WARNING 5: Be prepared to use grouping symbols whenever you perform a substitution; only omit them if you are sure you do not need them. Note that f ( 3) is not equal to 3 2, which equals 9.
(Section 1.1: Functions) 1.1.7 WARNING 6: As a matter of convenience, some sources refer to f ( x) as a function, but this convention is often rejected as non-rigorous. One advantage to the f x variable. The notation f x, y ( ) notation is that it indicates that f is a function of one ( ) indicates that f is a function of two variables. A function can be determined by a domain and a function rule. Together, the domain and the function rule determine the range of the function. (See Footnote 1.) Two functions with the same rule but different domains are considered to be different. Piecewise-defined functions will be discussed in Section 1.5. Such a function applies different rules to disjoint (non-overlapping) subsets of its domain (subdomains). For example, consider the function f, where: f ( x)= x2, 2 x < 1 x + 1, 1 x 2 PART E: POLYNOMIAL, RATIONAL, AND ALGEBRAIC FUNCTIONS Review Section 0.6 on polynomial, rational, and algebraic expressions. f is a polynomial function on f can be defined by: f ( x)= (a polynomial in x), which implies that Dom( f )=. See Footnote 2 on whether polynomial functions can be defined on another domain. x could be replaced by another variable. f is a rational function f can be defined by: f x ( )= (a rational expression in x). f is an algebraic function f can be defined by: f x ( )= (an algebraic expression in x). Example 9 (Polynomial, Rational, and Algebraic Functions) a) Let f ( x)= x 3 + 7x 5/7. Then, f is an algebraic function. 3 x x + 5 + 5t 3 1 b) Let g()= t. Then, g is both rational and algebraic. t 2 + 7t 2 c) Let h( x)= x 7 + x 2 3. Then, h is polynomial, rational, and algebraic.
(Section 1.1: Functions) 1.1.8 PART F: REPRESENTATIONS OF FUNCTIONS Ways to Represent a Function Rule The domain of the function could be the implied domain (see Part G). In Parts B and C, we determined the domain from a set of ordered pairs or an arrow diagram. For our examples in 1) through 8), we will let f be our squaring function from Example 8, with Dom f ( )=. A function rule can be represented by 1) a defining formula: 2) an input-output model (machine): f ( x)= x 2 x f x 2 3) a verbal description: This function squares its input, and the result is its output. 4) a table of input-output pairs: Input x Output f x ( ) 2 4 1 1 0 0 1 1 2 4 Since Dom f ( )=, a complete table is impossible to write. However, a partial table such as this can be useful, especially for graphing purposes.
5) a set of ( input x, output f ( x) ) ordered pairs: The table in 4) yields ordered pairs such as ( 2, 4). Since Dom( f )=, the set is an infinite set. (Section 1.1: Functions) 1.1.9 6) a graph consisting of points corresponding to the ordered pairs in 5); see Section 1.2: The graph of f below represents the set x, x 2 {( ) x }. We assume that the graph extends beyond the figure as expected. 7) an equation: The equations y = x 2 and y x 2 = 0 describe y as the same function of x (explicitly so in the first equation; implicitly in the second). Their common graph is in 6). See Section 1.2. 8) an arrow diagram: A partial arrow diagram for f is below. 9) an algorithm. Perhaps the output is computed by some computer code. 10) a series. (See Footnote 3.)
(Section 1.1: Functions) 1.1.10 PART G: IMPLIED (OR NATURAL) DOMAIN Implied (or Natural) Domain If f is defined by: f ( x)= (an expression in x), then the implied (or natural) domain of f is the set of all real numbers (x values) at which the value of the expression is a real number. x could be replaced by another variable. Dom( f ) is assumed to be the implied domain of f, unless otherwise specified or implied by the context. In some applications (including geometry), we may restrict inputs to nonnegative and/or integer values (rounding may be possible). Implied Domain of an Algebraic Function If a function is algebraic, then its implied domain is the set of all real numbers except those that lead to (the equivalent of) 1) dividing by zero, or 2) taking the even root of a negative-valued radicand. The list of restrictions will grow when we discuss non-algebraic functions. Example 10 (Implied Domain of an Algebraic Function) a) If f ( x)= 1 x all real numbers except 0. ( or x 1 ), then the implied domain of f is \{}, 0 the set of b) If g( x)= x ( or x 1/2 ), then the implied domain of g is 0, ), the set of all nonnegative real numbers. WARNING 7: The implied domain of g includes 0. Observe that 0 = 0, a perfectly good real number. WARNING 8: We will define 1, for example, as an imaginary number in Section 2.1. However, 1 will never be a real value.
(Section 1.1: Functions) 1.1.11 PART H: FINDING DOMAINS AND/OR RANGES Example 11 (Squaring Function: Finding Domain and Range) Let f ( x)= x 2. Describe the domain and the range of f using set-builder, graphical, and interval forms. x 2 is a polynomial, so we assume that Dom( f )=. The symbol is used in place of set-builder form. The graph of is the entire real number line: In interval form, is (, ). The resulting range of f is the set of all nonnegative real numbers, because every such number is the square of some real number. For example, 7 is the square of 7 : f ( 7 )= 7. Also: WARNING 9: Squares of real numbers are never negative. In set-builder form, the range is: { y y 0}, or { y : y 0}. (We could have used x instead of y, but we tend to associate y with outputs, and we should avoid confusion with the domain.) The graph of the range is: In interval form, the range is: 0, ).
(Section 1.1: Functions) 1.1.12 Example 12 (Finding a Domain) Let f ( x)= x 3. Find Dom( f ), the domain of f. x 3 is a real output x 3 0 x 3. WARNING 10: We solve the weak inequality x 3 0, not the strict inequality x 3 > 0. Observe that 0 = 0, a real number. The domain of f in set-builder form is: { x x 3}, or x : x 3 { } in graphical form is: in interval form is: 3, ) Range( f )= 0, ). Ranges will be easier to determine once we learn how to graph these functions in Section 1.4. If the rule for f had been given by f ()= t t 3, we still would have had the same function with the same domain and range. The domain could be written as { t t 3}, { x x 3}, etc. It s the same set of numbers. Example 13 (Finding a Domain) Let f ( x)= 1 x 3. Find Dom( f ). This is similar to Example 12, but we must avoid a zero denominator. We solve the strict inequality x 3 > 0, which gives us x > 3. The domain of f in set-builder form is: { x x > 3}, or { x : x > 3} in graphical form is: in interval form is: ( 3, )
(Section 1.1: Functions) 1.1.13 Example 14 (Finding a Domain) 4 Let f ( x)= 3 x. Find Dom( f ). Solve the weak inequality: 3 x 0. Method 1 Method 2 3 x 0 Now subtract 3 from both sides. x 3 Now multiply or divide both sides by 1. x 3 WARNING 11: We must then reverse the direction of the inequality symbol. 3 x 0 Now add x to both sides. 3 x Now switch the left side and the right side. x 3 WARNING 12: We must then reverse the direction of the inequality symbol. The domain of f in set-builder form is: { x x 3}, or x : x 3 in graphical form is: { } in interval form is: (,3 Example 15 (Finding a Domain) 3 Let f ( x)= x 3 Dom( f )=, because:. Find Dom( f ). The radicand, x 3, is a polynomial, and WARNING 13: The taking of odd roots (such as cube roots) does not impose any new restrictions on the domain. Remember that the cube root of a negative real number is a negative real number.
(Section 1.1: Functions) 1.1.14 Example 16 (Finding a Domain) Let g()= t 3t + 9 2t 2 + 20t. Find Dom( g). WARNING 14: Don t get too attached to f and x. Be flexible. The square root operation requires: 3t + 9 0 3t 9 t 3 We forbid zero denominators, so we also require: 2t 2 + 20t 0 2t( t+ 10) 0 t 0 and t + 10 0 t 0 and t 10 WARNING 15: We use the connective and here, not or. (See Footnote 4.) We already require t 3, so we can ignore the restriction t 10. The domain of g in set-builder form is: { t t 3 and t 0}, or { t : t 3 and t 0} in graphical form is: in interval form is: 3, 0) ( 0, ) (See Footnote 5 on our future study of domain and range.)
PART I: EVALUATING FUNCTIONS (THE MECHANICS) (Section 1.1: Functions) 1.1.15 In practice, we often evaluate a function at an input without finding the domain. 3t + 9 We immediately attempt to evaluate the defining expression, such as 2t 2 + 20t below, at the input. As we simplify, if we obtain an expression that is clearly undefined as a real value, we determine that the function value is undefined. Example 17 (Evaluating a Function; Revisiting Example 16) Let g()= t 3t + 9 2t 2 + 20t. Evaluate g (), 1 g ( ), g ( 0), and g ( 4). We write: g ()= 1 21 = 12 22 31 ()+ 9 () 2 + 20() 1 = 2 3 22 = 3 11 WARNING 16: Your answer must be in simplified form. g ( )= 3 + 9 2 2 + 20, or 3 + 3 ( ) ( ) 2 + 10 We also write (informally, as it turns out): g ( 0)= 20 = 9 0 g ( 4)= 30 ( )+ 9 ( ) 2 + 20( 0) 2 4 ( Undefined) 3( 4)+ 9 ( ) 2 + 20( 4) = 3 48 (Undefined as a real value) We saw in Example 16 that Dom( g)= 3, 0) ( 0, ). 1 and are in Dom( g), so g () 1 and g 0 and 4 are not in Dom( g), so g 0 ( ) are defined. ( ) and g ( 4) are undefined.
Example 18 (Evaluating a Function at a Non-numeric Input) Let f ( x)= 3x 2 2x + 5. Evaluate f ( x+ h). ( ) is often not equivalent to f x WARNING 17: f x+ h f ( x)+ f ( h). Instead, think: Substitution. (Section 1.1: Functions) 1.1.16 ( )+ h or To evaluate f ( x+ h), we take 3x 2 2x + 5, and we replace all occurrences of x with ( x + h). This may seem awkward, because we are replacing x with another expression containing x. f ( x)= 3x 2 2x + 5 f ( x+ h)= 3( x + h) 2 2( x + h)+ 5 = 3( x 2 + 2xh + h 2 ) 2x 2h + 5 WARNING 18: Be careful when squaring binomials and when applying the Distributive Property when an expression is being subtracted. = 3x 2 + 6xh + 3h 2 2x 2h + 5 We will see much more of the notation f x+ h ( ) when we cover difference quotients and derivatives in Sections 1.10, 1.11, and 5.7.
(Section 1.1: Functions) 1.1.17 PART J: APPLICATIONS In this chapter, we will discuss the following functions: Function s position or height (in Section 1.2) Input t = the time elapsed (in seconds) after a coin is dropped from the top of a building Output (Function Value) st ()= the height (in feet) of the coin t seconds after it is dropped f temperature conversion (in Section 1.9) x = the temperature using the Celsius scale f ( x)= the Fahrenheit equivalent of x degrees Celsius P profit (in Sections 1.10, 1.11) x = the number of widgets produced and sold by a company P x ( )= the resulting profit when x widgets are produced and sold
(Section 1.1: Functions) 1.1.18 FOOTNOTES 1. Terminology. When defining a function f, some sources require: a domain (a set containing all of the inputs in the ordered pairs making up the function, but nothing else), Some sources attempt to define the domain and the range of a relation, but there is disagreement as to how to define the domain (and also the range, as discussed below). Some sources allow the domain to include elements that are not inputs for any of the ordered pairs in the relation. a codomain (a set containing all of the outputs and perhaps other elements that are not outputs), and a function rule, perhaps obtained from the statement of f as a set of ordered pairs, relating each (input) element of the domain to exactly one (output) element of the codomain. We can then write f : X Y, meaning that f maps the domain X to the codomain Y. Let f ( x)= x 2, also denoted by f : x x 2, where the domain is and the codomain is. We can then write f :. This is because f relates each real number input in the domain to exactly one real number output, which is an element of the codomain. The range, which is the set of all assigned outputs, is a subset of the codomain. In the example above, the range is a proper subset of the codomain, because not every real number in the codomain is assigned. Specifically, the negative reals are not assigned. What we call the codomain some sources call the range, and what we call the range some authors call the image of the function. 2. Definition of a polynomial function. If f ( x)= x2 x, then f ( x )= x ( x 0). Is f a polynomial function? In his book Polynomials (New York: Springer-Verlag, 1989), E.J. Barbeau implies that it is. Other sources imply otherwise due to the fact that Dom( f ). It depends on whether or not one views Dom( f )= as a defining characteristic of a polynomial function f for now. In Chapter 2, we will see cases where Dom( f )=.
(Section 1.1: Functions) 1.1.19 3. Series expansions of [defining expressions of] functions. Let f ( x)= 1. In Section 9.4 1 x and in calculus, you will see that f ( x) has the infinite series expansion 1+ x + x 2 + x 3 +..., provided that 1 < x < 1. In calculus, you will consider series expansions for sin x, cos x, e x, etc. 4. The Zero Factor Property and inequalities. According to the Zero Factor Property, if ab = 0 for real numbers a and b, then a = 0 or b = 0. If we were solving the equation 2t( t+ 10)= 0, we could use the Zero Factor Property. 2t( t+ 10)= 0 ( t = 0) or ( t = 10) In Example 16, we essentially solved the inequality 2t( t+ 10) 0. ~ denotes negation ( not ). 2t( t+ 10) 0 ~ ( t = 0) or ( t = 10) ~ ( t = 0) and ~ ( t = 10) by DeMorgan's Laws of logic (see below) ( ) and ( t 10) t 0 By DeMorgan s Laws of logic, ~ p or q. For example: If I am an American, then (I am an Alabaman) or (I am an Alaskan) or. If I am not an American, then (I am not an Alabaman) and (I am not an Alaskan) and. ( ) is logically equivalent to ( ~ p) and ( ~ q) A Zero Factor Property for inequalities: If ab 0 for real numbers a and b, then a 0 and b 0. 5. Revisiting domain and range. In Section 1.2, we will relate domains and ranges to graphs. We will study domains and ranges of basic functions in Section 1.3; more complicated functions in Sections 1.4, 1.5, and 1.6; inverse functions in Section 1.9 (and Section 4.10); and various types of functions in Chapters 2, 3, and 4. The topic of solving nonlinear inequalities in Section 2.11 will be relevant, particularly when finding domains of algebraic functions. In Section 9.2, we will study sequences, which are functions with domains consisting of only integers. Ranges will be better understood as we discuss graphs in further detail.
(Section 1.2: Graphs of Functions) 1.2.1 SECTION 1.2: GRAPHS OF FUNCTIONS LEARNING OBJECTIVES Know how to graph a function. Recognize when a curve or an equation describes y as a function of x, and apply the Vertical Line Test (VLT) for this purpose. Recognize when an equation describes a function explicitly or implicitly. Use a graph to estimate a function s domain, range, and specific function values. Find zeros of a function, and relate them to x-intercepts of its graph. Use a graph to determine where a function is increasing, decreasing, or constant. PART A: DISCUSSION In Chapter 0, we graphed lines and circles in the Cartesian plane. If f is a function, then its graph in the usual Cartesian xy-plane is the graph of the equation y = f ( x), and it must pass the Vertical Line Test (VLT). In Section 1.8, we will consider the graph of x = f ( y). In this section, we will sketch graphs of functions. We will investigate how their behaviors reflect the behaviors of their underlying functions, as well as the information that they contain about those functions. For example, the real zeros of a function f correspond to the x-intercepts of its graph in the xy-plane. If it exists, f ( 0) gives us the y-intercept. Also, a function increases and decreases according to the rising and falling of its graph. After this section, we will specialize and focus on particular functions and categories of functions, as well as their corresponding graphs.
(Section 1.2: Graphs of Functions) 1.2.2 PART B: THE GRAPH OF A FUNCTION The graph of a function f in the xy-plane is the graph of the equation y = f ( x). It consists of all points of the form ( x, f ( x) ), where x Dom( f ). In Example 13, we will graph the function s in the th-plane by graphing h = st (). Remember that, as a set of ordered pairs, f = x, f ( x) ( ) Here, as we typically assume, {( ) x Dom f }. x is the independent variable, because it is the input variable. y is the dependent variable, because it is the output variable. Its value (the function value) typically depends on the value of the input x. Then, it is customary to say that y is a function of x, even though y is a variable here and not a function. The form y = f x ( ) implies this. In Section 1.8, we will switch the roles of x and y. Basic graphs, such as the ones presented in Section 1.3, and methods of manipulating them, such as the ones presented in Section 1.4, are to be remembered. The Point-Plotting Method presented below will help us develop basic graphs, and it can be used to refine our graphs by identifying particular points on them. It is also available as a last resort if memory fails us. The Point-Plotting Method for Graphing a Function f in the xy-plane Choose several x values in Dom( f ). For each chosen x value, find f ( x), its corresponding function value. Plot the corresponding points x, f x ( ( )) in the xy-plane. Try to interpolate (connect the points, though often not with line segments) and extrapolate (go beyond the scope of the points) as necessary, ideally based on some apparent pattern. Ensure that the set of x-coordinates of the points on the graph is, in fact, Dom f ( ).
PART C: GRAPHING A SQUARE ROOT FUNCTION Let f ( x)= x. We will sketch the graph of f in the xy-plane. This is the graph of the equation y = f ( x), or y = x. (Section 1.2: Graphs of Functions) 1.2.3 TIP 1: As usual, we associate y-coordinates with function values. When point-plotting, observe that: Dom( f )= 0, ). For instance, if we choose x = 9, we find that f ( 9)= 9 = 3, which means that the point ( 9, f ( 9) ), or ( 9, 3), lies on the graph. On the other hand, f ( 9) is undefined, because 9 Dom( f ). Therefore, there is no corresponding point on the graph with x = 9. A (partial) table can help: Below, we sketch the graph of f : x ( ) Point ( ) ( ) ( ) ( ) f x 0 0 0, 0 1 1 1, 1 4 2 4, 2 9 3 9, 3 WARNING 1: Clearly indicate any endpoints on a graph, such as the origin here. The lack of a clearly indicated right endpoint on our sketch implies that the graph extends beyond the edge of our figure. We want to draw graphs in such a way that these extensions are as one would expect. WARNING 2: Sketches of graphs produced by graphing utilities might not extend as expected. The user must still understand the math involved. Point-plotting may be insufficient. The x between the 4 and the 9 on the x-axis represents a generic x-coordinate in Dom f ( ). We could use x 0 ( x sub zero or x naught ) to represent a particular or fixed x-coordinate.
PART D: THE VERTICAL LINE TEST (VLT) (Section 1.2: Graphs of Functions) 1.2.4 The Vertical Line Test (VLT) A curve in a coordinate plane passes the Vertical Line Test (VLT) There is no vertical line that intersects the curve more than once. An equation in x and y describes y as a function of x Its graph in the xy-plane passes the VLT. Then, there is no input x that yields more than one output y. Then, we can write y = f ( x), where f is a function. A curve could be a straight line. Example 1 (Square Root Function and the VLT; Revisiting Part C) The equation y = x explicitly describes y as a function of x, since it is of the form y = f ( x). f is the square root function from Part C. Observe that the graph of y = x passes the VLT. Each vertical line in the xy-plane either misses the graph entirely, meaning that the corresponding x value is not in Dom f ( ), or intersects the graph in exactly one point, meaning that the corresponding x value yields exactly one y value as its output.
Example 2 (An Equation that Does Not Describe a Function) (Section 1.2: Graphs of Functions) 1.2.5 Show that the equation x 2 + y 2 = 9 does not describe y as a function of x. (Method 1: VLT) The circular graph of x 2 + y 2 = 9 below fails the VLT, because there exists a vertical line that intersects the graph more than once. For example, we can take the red line ( x = 2) below: Therefore, x 2 + y 2 = 9 does not describe y as a function of x. (Method 2: Solve for y) This is also evident if we solve x 2 + y 2 = 9 for y: x 2 + y 2 = 9 y 2 = 9 x 2 y =± 9 x 2 Any input value for x in the interval ( 3, 3) yields two different y outputs. For example, x = 2 yields the outputs y = 5 and y = 5.
(Section 1.2: Graphs of Functions) 1.2.6 PART E: IMPLICIT FUNCTIONS and CIRCLES Example 3 (An Equation that Describes a Function Implicitly) The equation xy = 1 implicitly describes y as a function of x. This is because, if we solve the equation for y, we obtain: y = 1 x. This is of the form y = f ( x), where f is the reciprocal function. Example 4 (Implicit Functions and Circles; Revisiting Example 2) As it stands, the equation x 2 + y 2 = 9 does not describe y as a function of x; we saw this in Example 2. However, it does provide implicit functions if we impose restrictions on x and/or y and consider smaller pieces of its graph. If we impose the restriction y 0 and solve the equation x 2 + y 2 = 9 for y, we obtain y = 9 x 2. (See Example 2.) Its graph is the upper half of the circle, and it passes the VLT, so y = 9 x 2 does describe y as a function of x. If we impose the restriction y 0 and solve the equation x 2 + y 2 = 9 for y, we obtain y = 9 x 2. Its graph is the lower half of the circle, and it passes the VLT, so y = 9 x 2 does describe y as a function of x. This helps us graph entire circles on graphing utilities.
(Section 1.2: Graphs of Functions) 1.2.7 PART F: ESTIMATING DOMAIN, RANGE, and FUNCTION VALUES FROM A GRAPH The domain of f is the set of all x-coordinates of points on the graph of y = f x ( ). (Think of projecting the graph onto the x-axis.) The range of f is the set of all y-coordinates of points on the graph of y = f x ( ). (Think of projecting the graph onto the y-axis.) Domain Think: x f Range Think: y Example 5 (Estimating Domain, Range, and Function Values from a Graph) Let f ( x)= x 2 + 1. Estimate the domain and the range of f based on its graph below. Also, estimate f (). 1 Apparently, Dom( f )=, or (, ), and Range( f )= 1, ). It also appears that the point ( 1, 2) lies on the graph and thus f ()= 1 2. Finding the range of a function will become easier as you learn how to graph functions in precalculus and calculus. WARNING 3: Graph analyses can be imprecise. The point 1, 2.001 ( ), for example, may be hard to identify on a graph. Not all coordinates are integers.
PART G: ZEROS (OR ROOTS) and INTERCEPTS (Section 1.2: Graphs of Functions) 1.2.8 The real zeros (or roots) of f are the real solutions of f ( x)= 0, if any. They correspond to the x-intercepts of the graph of y = f ( x). WARNING 4: The number 0 may or may not be a zero of f. In this sense, the term zero may be confusing. On the other hand, the term root might be confused with square roots and such. The graph of y = f x infinitely many, depending on f. ( ) can have any number of x-intercepts (possibly none), or We typically focus on real zeros, though we will discuss imaginary zeros in Chapters 2 and 6. The y-intercept of the graph of y = f ( x), if it exists, is given by f ( 0) or by the point ( 0, f ( 0) ). The graph of y = f x ( ) can have at most one y-intercept. Example 6 (Finding Zeros and Intercepts) Find the zeros (or roots) of f, where f ( x)= x 2 9, and find the x-intercepts of the graph of y = f ( x). Solve f ( x)= 0: x 2 9 = 0 x 2 = 9 x =±3 The zeros of f are 3 and 3. They are both real, so they correspond to x-intercepts of the graph of y = x 2 9. Some prefer to write the x-intercepts as ( 3, 0) and ( 3, 0). WARNING 5: Do not confuse the process of finding zeros, which involves solving the equation f ( x)= 0, with the process of evaluating at 0, which involves substituting ( 0) for x and finding f ( 0). Here, f ( 0)= 9. In fact, 9, or the point ( 0, 9), is the y-intercept.
(Section 1.2: Graphs of Functions) 1.2.9 The graph of f is below. We will informally refer to zeros of the defining expression for a function, in particular zeros of radicals and fractions. Zeros of a Radical n g( x) = 0 g( x)= 0 ( n = 2, 3, 4,...) That is, the zeros of a radical are the zeros of its radicand. Example 7 (Finding Zeros of a Radical ) Find the zeros (or roots) of f, where f ( x)= x 2 9. The zeros are the same as those for x 2 9, namely 3 and 3. The graph of f is below. Why does the graph disappear on the x-interval ( 3, 3)?
(Section 1.2: Graphs of Functions) 1.2.10 Zeros of a Fraction If f ( x) is of the form numerator N x denominator D x ( ) ( ), then the zeros of f are the zeros of N that are in Dom( f ) (WARNING 6). In particular, a zero of f cannot make any denominator undefined or equal to 0. Example 8 (Finding Zeros of a Fraction ) Find the zeros (or roots) of f, where f ( x)= x2 9 x + 7. Solve f ( x)= 0: x 2 9 x + 7 = 0 x 2 9 = 0 ( x 7) Again, the zeros of f are 3 and 3. The graph of f here has features we will discuss in Chapter 2. Example 9 (Finding Zeros of a Fraction ) If f ( x)= x2 9, the only zero of f is 3, because 3 is not in x 3 Dom( f ). (3 yields a zero denominator.)
(Section 1.2: Graphs of Functions) 1.2.11 Example 10 (Finding Zeros) Find the zeros (or roots) of g, where g()= t 3t 2 t 4. t 3 Observe that 3 is excluded from Dom( g), because it yields a zero denominator. Dom( g) also excludes values of t that yield negative values for the radicand, 3t 2 t 4. We don t have to worry about this, though, because we only care about values of t that make that radicand zero in value, anyway. Solve g()= t 0 : Method 1: Factoring 3t 2 t 4 = 0 t 3 3t 2 t 4 = 0 t 3 ( ) ( ) 3t 2 t 4 = 0 t 3 ( 3t 4) ( t + 1)= 0 ( t 3) By the Zero Factor Property, 3t 4 = 0 t = 4 3 or t + 1 = 0 t = 1 WARNING 7: If we had obtained 3, we would have had to eliminate it. The zeros of g are 4 3 and 1.
Method 2: Quadratic Formula We need to solve: 3t 2 t 4 = 0 ( t 3). (Section 1.2: Graphs of Functions) 1.2.12 For the Quadratic Formula, a = 3, b = 1, and c = 4. WARNING 8: It helps to identify what a, b, and c are. Sign mistakes are common. Apply the Quadratic Formula. t = b ± b2 4ac 2a = 1 ( )± 1 = 1± 1+ 48 6 1± 49 = 6 = 1± 7 6 ( ) 2 43 ( )( 4) 23 ( ) Using + : Using : t = 1+ 7 6 = 8 6 = 4 3 t = 1 7 6 = 6 6 = 1 WARNING 9: Again, if we had obtained 3, we would have had to eliminate it. Again, the zeros of g are 4 3 and 1.
(Section 1.2: Graphs of Functions) 1.2.13 PART H: INTERVALS OF INCREASE, DECREASE, AND CONSTANT VALUE We may have an intuitive sense of what it means for a function to increase (respectively, decrease, or stay constant) on an interval. In Examples 11 and 12, we will formalize this intuition. Example 11 (Intervals of Increase and Intervals of Decrease from a Graph) Let f( x)= x 3 3x + 2. The graph of f is below. Give the intervals of increase and the intervals of decrease for f. It is assumed that we give the largest intervals in the sense that no interval we give is a proper subset of another appropriate interval. f increases on the interval (, 1. Why? Graphically: If we only consider the part of the graph on the x-interval (, 1, any point must be higher than any point to its left. The graph rises from left to right. Numerically: Any x-value in the interval (, 1 yields a greater function value f x ( ) than any lesser x-value in the interval does. f increases on an interval I x 2 > x 1 implies that f ( x 2 )> f ( x 1 ), x 1, x 2 I.
f decreases on the interval 1, 1. Why? (Section 1.2: Graphs of Functions) 1.2.14 Graphically: If we only consider the part of the graph on the x-interval 1, 1, any point must be lower than any point to its left. The graph falls from left to right. Numerically: Any x-value in the interval 1, 1 yields a lesser function value f x ( ) than any lesser x-value in the interval does. f decreases on an interval I x 2 > x 1 implies that f ( x 2 )< f ( x 1 ), x 1, x 2 I. f increases on the interval 1, ). Example 12 (Intervals of Constant Value from a Graph) The graph of g below implies that g is constant on the interval 1, 1, because the graph is flat there. f is constant on an interval I f ( x 2 )= f ( x 1 ), x 1, x 2 I. In calculus, you will reverse this process. You will first determine intervals where a function is increasing / decreasing / constant, and then you will sketch a graph. You will locate turning points such as the ones indicated on the graph of f in Example 11. The point 1, 4 The point 1, 0 ( ) is called a local (or relative) maximum point. ( ) is called a local (or relative) minimum point. Derivatives, which are key tools, will be previewed in Section 1.11.
(Section 1.2: Graphs of Functions) 1.2.15 PART I: USING OTHER NOTATION WARNING 10: Don t get too attached to y, f, and x. Be flexible. Example 13 (Falling Coin) You drop a coin from the top of a building. Let t be the time elapsed (in seconds) since you dropped the coin. Let h be the height (in feet) of the coin. Let s be a position function such that h = st (). We ignore what happens after the coin hits the ground. Instead of graphing y = f ( x), we graph h = st (). t, not x, is the independent variable. h, not y, is the dependent variable. s, not f, is the function. the th-plane, not the xy-plane, is the coordinate plane containing the graph of the function s. The graph of s, or the graph of h = st (), in the th-plane is given below. As a set of ordered pairs, s = ( t, st ()) t Dom() s { }. WARNING 11: The horizontal and vertical axes are scaled differently here. We typically try to avoid this unless necessary. The reader can analyze this graph, including the indicated points, in the Exercises.