A Basic Toolbox of Functions S. F. Ellermeyer June 1, 3 Abstract We introduce some basic functions that are commonly used in mathematical investigations. You should become familiar with these functions and their properties. 1 The Identity Function The identity function is the function f = {(x, y) y = x} or simply f (x) =x with domain (f) =(, ). Thegraphoff is pictured below. 4-4 - x 4 - -4 Graph of f (x) =x 1
It is easy to observe that the range of f is range (f) =(, ). The identity function is a simple example of a linear function. A linear function is any function of the form f (x) =mx + b where m and b are constants. (For the identity function, we have m =1and b =.) We will embark on a more general study of linear functions later in this course. Constant Functions A constant function, f, isafunctionwithdomain (f) =(, ) all of whose pairs have the same second component. If K (a constant) is the value of this second component, then range (f) ={K}. An example of a constant function is f (x) = 4. The graph of this function is shown below. 8 6 4-4 - x 4 - -4-6 -8 Graph of f (x) = 4 Note that a constant function, f (x) =K is a linear function because it has the form f (x) =mx + b with m =and b = K. 3 The Squaring Function The squaring function is the function f = (x, y) y = x ª
or simply f (x) =x with domain (f) =(, ). The graph of f, which is shown below, is a parabola with vertex at the point (, ). The range of f is range (f) =[, ). 15 1 5 and -4 - x 4 Graph of f (x) =x Observe that if x is any real number, then Thus, if x is any real number, then f (x) =x f ( x) =( x) =( x)( x) =x. f ( x) =f (x). Any function for which f ( x) =f (x) for all x domain (f) is called an even function. Iff is any even function and the pair (x, y) is a member of f, then the pair ( x, y) isalsoamemberoff. The squaring function is the simplest example of a quadratic function. A quadratic function is any function of the form f (x) =ax + bx + c where a, b, and c are constants with a 6=. (For the squaring function, we have a =1,b =,and c =). Every quadratic function has a parabolic graph. We will study quadratic functions in detail later in this course. 3
If we like (perhaps because it is called for in some applied problem), we can restrict the domain of the squaring function. An example of a squaring function with restricted domain is The graph of g is shown below. g (x) =x,x [ 3, ]. 14 1 1 8 6 4-4 -3 - -1 1 x 3 4 Graph of g (x) =x,x [ 3, ] Note that if f is any function, we can always restrict the domain of f to obtain a new function, g, whose graph is a proper subset of the graph of f. We will not comment further on this idea in the remainder of our introduction to a basic toolbox of functions, but it should be kept in mind that restriction of a domain is something that can always be done if desired. 4 The Cubing Function The cubing function is the function or simply f = (x, y) y = x 3ª f (x) =x 3 with domain domain (f) =(, ). The graph of f is shown below. The range of f is range (f) =(, ). 4
1 5-4 - x 4-5 -1 and Graph of f (x) =x 3 Observe that if x is any real number, then f (x) =x 3 f ( x) =( x) 3 =( x)( x)( x) = x 3. Thus, if x is any real number, then f ( x) = f (x). Any function for which f ( x) = f (x) for all x domain (f) is called an odd function. If f is any odd function and the pair (x, y) is a member of f, then the pair ( x, y) isalsoamemberoff. 5 The Square Root Function If x is any non negative real number, then the square root of x, denoted by x, is the unique non negative real number, y, such that y = x. Thus, when we write y = x, we mean that y = x and y. Note that for every positive number, x, there are actually two different real numbers, y, such that y = x. Forexample, =4and ( ) =4.However, 5
it is not correct to say that 4= because our definition asserts that 4 must not be negative. Thus 4=. Thesquarerootfunctionisthefunction f = (x, y) y = x ª or simply f (x) = x with domain domain (f) =[, ). The graph of f is shown below. The range of f is range (f) =[, ). 3 y 1 4 6 8 1 1 14 16 x Graph of f (x) = x Note that the square root function is neither even nor odd. The concepts of even and odd apply only to functions, f, forwhichthenumber x is in domain (f) for every number x in domain (f). 6 The Absolute Value Function If x is any real number, then the absolute value of x, denotedby x, is defined as follows: ½ x if x x = x if x<. Thus, for example, since 1, wehave 1 =1, 6
and since 8 <, wehave 8 = ( 8) = 8. Some important observation about the absolute value are the following: 1. If x is any real number, then x.. If x is any real number, then x is the distance on the number line between x and. (This is illustrated in the figure below.) Theabsolutevaluefunctionisthefunction or simply f = {(x, y) y = x } f (x) = x with domain (f) =(, ). Thegraphoff is shown below. The range of f is range (f) =[, ). 4 3 1-4 - x 4 Graph of f (x) = x 7
7 The Reciprocal Function If x is any non zero real number, then the reciprocal of x is defined to be 1/x. For example, the reciprocal of is 1 and the reciprocal of 1 6 is 1 1 = 6 The reciprocal function is the function f = 1 1 1 6 = ½ (x, y) y = 1 ¾ x µ µ 1 6 = 6. 1 1 or simply f (x) = 1 x with domain (, ) (, ). The graph of f is shown below. The range of f is (, ) (, ). 1 8 6 4-4 - - x 4-4 -6-8 -1 Graph of f (x) = 1 x A noteworthy feature of the graph of the reciprocal function (not seen on any of the graphs of the basic functions we have studied so far) is that the graph has two asymptotes. One is a vertical asymptote and the other is a horizontal asymptote. 8
In order to understand the concept of asymptote, observe that very large values of x correspond to very small values of y. For example, f (1) = 1 1 f (1, ) = 1 1, f ( 47, ) = 1 47,. Note that we can make the y value be as small (that is, as close to ) as we like by choosing x large enough (meaning large in absolute value and either positive or negative depending on whether we want y to be positive or negative). We describe this phenomenon by writing lim y = x lim y = x The firstoftheabovestatements isread thelimit as x approaches infinity of y equals, and means that we can make y be as close as we like to by choosing x positive and large enough. The second statement is read the limit as x approaches negative infinity of y equals, and means that we can make y beascloseasweliketo by choosing x negative and large enough (in absolute value). Either of the above two observations implies that the horizontal line y =is a horizontal asymptote of the graph of f. Next, observe what happens when we choose very small (meaning very close to ) valuesofx: µ 1 f =1 f µ 1 1, 1 = 1, f (.376) = 1, 66..376 If x is very small and positive, then y is very large and positive. If x is very small and negative, then y is very large (in absolute value) and negative. We 9
describe this by writing lim y = x + lim y = x The first of these statements is read the limit as x approaches from the right of y is infinity and means that y canbemadeaslarge(andpositive) as we like by choosing x small enough (and positive). The second statement is read the limit as x approaches fromtheleftofy is negative infinity and means that y can be made as large in absolute value (and negative) as we like by choosing x small enough (and negative). Either of the above two observations implies that the vertical line x =is a vertical asymptote of the graph of f. In general, if f is any function (with independent variable x and dependent variable y) whose domain contains an infinite interval and if M is a constant such that either lim f (x) =M or lim f (x) =M, x x then the horizontal line y = M is said to be a horizontal asymptote of the graph of f. Likewise, if f is any function (with independent variable x and dependent variable y) whose domain contains a finite interval, (a, b), with the possible exception of a single number m (a, b) and if any one of the conditions lim f (x) =, lim x m + f (x) =, lim x m f (x) =, lim x m + f (x) = x m is fulfilled, then the vertical line x = m is said to be a vertical asymptote of the graph of f. Exercise 1 1. Is the identity function even, odd, or neither?. Is a constant function even, odd, or neither? 3. Is the square root function even, odd, or neither? 4. Is the absolute value function even, odd, or neither? 5. Is the reciprocal function even, odd, or neither? 1
6. Is it possible for a function to be both even and odd? Explain your answer. 7. Graphs of three functions are shown below. Decide whether each of these functions is even, odd, or neither. -4-4 x Graph of a function, f -4-4 x Graph of a function, g 11
14 1 1 8 6 4 - -1 1 3x 4 5 6 Graph of a function, h 8. Sketch the graphs of the following functions with restricted domains. For each, state the range. (a) f (x) =x, x [, 3] (b) f (x) =6, x ( 4, 1) (c) f (x) =x, x [, 3) (d) f (x) =x 3, x ( 3, ) (e) f (x) = x, x [, 64] (f) f (x) = x, x [, ) (g) f (x) = 1, x [ 1, ) x 1 9. Which horizontal lines appear to be horizontal asymptotes of the graph shown below? Does the graph of this function appear to have any vertical asymptotes? 3 1-1 -8-6 -4-4 x 6 8 1-1 - -3 1
1. Find all horizontal and vertical asymptotes of the function whose graph is shown below. (Hint: The domain of this function is all real numbers except for 1 and.) 15 1 5-5 -4-3 - -1 1 x 3 4 5-5 -1-15 - 11. If f is any function, then no two ordered pairs in f have the same first component. In other words, if (x 1,y 1 ) f and (x,y ) f, thenx 1 6= x. (This is the essence of our definition of the concept of function.) However, some functions contain ordered pairs that do have the same second component. For example the squaring function, f (x) = x, contains both of the pairs (, 4) and (, 4). A function, f, forwhich no two ordered pairs have the same second component is called a one to one function. Which of the following functions are one to one functions? (a) f (x) =x (b) f (x) = (c) f (x) =x (d) f (x) =x 3 (e) f (x) = x (f) f (x) = x (g) f (x) = 1 x (h) f (x) =x, x (i) f (x) =x, x 5 13
(j) f = {(, ), (, 4), (3, 8), (4, 1), (5, )} (k) f = {(5, 5), (1, 4), (3, 3), (4, 1)} (l) f = {(x, y) x + y =1and y }. 1. If you are shown the graph of a function, how can you tell whether or not the function is one to one? 14