Definition of a function. Elementary Functions. A function machine. Inputs and unique outputs of a function. Sam Houston State University

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1 We study the most fundamental concept in mathematics, that of a In this lecture we first define a function and then examine the domain of functions defined as equations involving real numbers. Part, Functions Lecture.a, The Definition of a Function. A function f : X Y assigns to each element of the set X an element of Y. Picture a function as a machine, Dr. Ken W. Smith Sam Houston State University / 7 A function machine We study the most fundamental concept in mathematics, that of a In this lecture we first define a function and then examine the domain of functions defined as equations involving real numbers. A function f : X Y assigns to each element of the set X an element of Y. Picture a function as a machine, / 7 Inputs and unique outputs of a function The set X of inputs is called the domain of the function f. The set Y of all conceivable outputs is the codomain of the function f. The set of all outputs is the range of f. (The range is a subset of Y.) The most important criteria for a function is this: A function must assign to each input a unique output. We cannot allow several different outputs to correspond to an input. dropping x-values into one endelementary of the Functions machine and picking up y-values at 3 / 7 4 / 7

2 Examples of functions Not a function We give an example (from Wikipedia) of a function from a set X to the set Y. The function maps to D, to C and 3 to C. Note that each element of X has a unique output in Y. However the map below is not a Some items in X are not mapped anywhere; worse, the item has two outputs, both B and C. Functions are not allowed to change a single input into several outputs! 5 / 7 Functions as questions Functions occur naturally in our world. When we pull out an attribute of an object, we are essentially creating a For example, the set X below has polygons with various colors. The question, What is the color of a polygon? could be viewed as a function that maps to polygons to colors. 6 / 7 SSN and Sam ID as functions Functions occur throughout our modern technological society. The US social security number is a function SSN mapping US citizens to nine digit numbers. At Sam Houston State University, all students and staff are assigned a Sam ID. This as a function SamID, mapping students/staff to nine digit numbers. For example, SamID(Ken W Smith) = (This function exists so that data about students/staff classes, grades, wages, etc. can be kept in a computer database, tracked by a single number.) 7 / 7 8 / 7

3 Functions as ordered pairs A worked exercise Although functions in science are often defined by equations, they do not have to be. (The SamID function is not defined by an equation.) In its most general form, a function is a collection of ordered pairs satisfying certain requirements. Consider the sets D := {, a, b, z, orange} and C := {r, s, t, u, v, 000}. We create a function f by assigning to each member of D a member of C. input output r a s b r z 000 orange 000 This is a function: the domain is the elements of D. And each element of D has a unique output! We may sometimes define a function by a table or by a list of ordered pairs. f = {(, r), (a, s), (b, r), (z, 000), (orange, 000)} 9 / 7 (The ordered pairs are simply the entries in the table.) Example # Worked Exercise. Consider the function with domain D = {,, 0,, }, codomain the real numbers R, defined by the formula g(x) = x. Display the function g in tabular form, and Display the function g as a set of ordered pairs. 3 Give the range of the function g. Solution. As a table, we can write out the function g as x g(x) As a set of order pairs, g = {(, 4), (, ), (0, 0), (, ), (, 4)} 3 The range of the function g is {0,, 4}. 0 / 7 Another example. Consider the function defined earlier. In the next lecture we examine functions defined by equations. Write this function in both tabular form and as a set of ordered pairs. Solution. In tabular form we have: X Y D C 3 C As ordered pairs, the function is the set {(, D), (, C), (3, C)}. / 7 (END) / 7

4 Functions defined by equations Many functions we explore in mathematics and science are defined by an equation. We can define a function implicitly in an equation involving two variables. For example, does the equation Part, Functions Lecture.b, Functions defined by equations x + 3y 4 = 0 define a function with inputs x and outputs y? Isolate y to get Dr. Ken W. Smith 3y = 4 x and so y = (4 x). 3 Sam Houston State University We may now explicitly define the function f (x) = (4 x). 3 So YES, the equation x + 3y 4 = 0 does define a 3 / 7 Independent and dependent variables 4 / 7 Exercises on implicit functions A digression. When we considered the equation x + 3y 4 = 0 our choice of x as input and y as output is arbitrary. We could decide (contrary to custom!) that y is the input and x is the output. Then, solving for x, we have x = 4 3y and so x = (4 3y) and so we create the function g(y) = (4 3y). But most of the time we will stick to convention and, unless stated otherwise, assume x is the input variable and y is the output variable. The input variable x is often called the independent variable and the output variable is the dependent variable since its value depends on the input. 5 / 7 Some worked exercises. Does the equation x y = 4 define y as a function of x? (If it does, give the domain of the implied ) Solution. We attempt to solve for y. We may multiply both sides of 4 the equation by as long as x is not zero. This gives us y =. x x Is there a problem with x = 0? No, x = 0 does not allow x y = 4, so x will never be zero in this equation. 4 Answer: YES, this is a function; y =. x The domain of this function is all real numbers except zero. In interval notation the domain is (, 0) (0, ). 6 / 7

5 Not a function Not a function Does the equation xy = 4 define y as a function of x? Solution. If we attempt to solve for y, we multiply both sides of the equation by x (as long as x 0) and so we have y = 4 x. But now, what is y? y could be positive or negative there will generally be two choices here, one positive and one negative. The appearance of two answers violates the uniqueness requirement in our outputs for a 3 Does the equation x y = 0 define y as a function of x? (Why/why not?) Solution. Although it might be tempting to solve for y, first notice that if x is zero then y could be 0 or or.788 or anything! So the input x = 0 does not give a unique output. This is not a Answer: NO; if x = 0 then y could be anything. Answer: NO, this is not a If x = then we don t know if y = or y =. (This is different than problem. In problem, x = 0 is not a possible input in the equation. But here x = 0 is a possibility for a solution to the equation! So we have to worry about the input x.) 7 / 7 8 / 7 Practicing function notation Let us practice the function notation, f(x). A formula for f(x) tells us how the input x leads to the output f(x). For example, suppose f(x) = x 9. Compute: f(0), f(), 3 f( ), 4 f( 5), 5 f( x) Solutions. If f(x) = x 9 then f(0) = 0 9 = 9. f() = () 9 = 9 = 8. 3 f( ) = ( ) 9 = 9 = 8, 4 f( 5) = ( 5) 9 = 5 9 = 6. 5 f( x) = ( x) 9 = x 9. 9 / 7 Practicing function notation More examples. Let s continue with the function f(x) = x 9. Compute: 6 f(x + h), 7 f( x), 8 f(a + ), 9 f(x) + Solutions. If f(x) = x 9 then 6 f(x + h) = (x + h) 9 = (x + xh + h ) 9 = x + xh + h 9, 7 f( x) =( x) 9 = x 9, 8 f(a + ) = (a + ) 9 = (4a + 4a + ) 9 = 4a + 4a 8, 9 f(x) + = (x 9) + = x +. 0 / 7

6 Function notation Part, Functions Lecture.c, Finding the domains of functions In the next presentation we find the domains of functions. (END) Dr. Ken W. Smith Sam Houston State University / 7 Domains of functions The domain of a function is (generally) the largest possible set of inputs into the Let s find the domain of the function f (x) = x. Example. Find the domain of the function x 3 g(x) = + + x 5. x + x + It is often easier to ask the question, What is not in the domain?. For the function f (x) = x we ask the question, Which real numbers do not have a square root? We cannot evaluate f (x) at negative numbers since the square of a real number cannot be negative. So the domain must be numbers which are not negative, that is, zero and positive real numbers. (We can indeed take the square root of 0 so we want to include 0 in the domain.) We can write our answer in interval notation: Solution. The domain of f (x) = x is [0, ). 3 / 7 / 7 Solution. What numbers cannot serve as input to g(x)? Since we cannot have denominators equal to zero then x = cannot be an input; neither can x =. So the domain of this function g is all real numbers except x = and x =. There are several ways to write the domain of g. Using set notation, we could write the domain as {x R : x 6=, }. This is a precise symbolic way to say, All real numbers except and. We could also write the domain in interval notation: (, ) (, ) (, ). 4 / 7 This notation says that the domain includes all the real numbers smaller

7 Some worked exercises. Find the domain of the function f(x) = x Solution. Since the square root function requires nonnegative inputs, we must have x 0. Therefore we must have x. The domain is [, ). Find the domain of the function f(x) = x x 3 Solution. Again, we must have x but we must also prevent the denominator from being zero, so x cannot be 3, either. The domain is then all real numbers at least as big as except for the number 3. Here is our answer in interval notation: The domain is [, 3) (3, ). 5 / 7 6 / 7 3 Find the domain of the function f(x) = x x 6x + 8 Solution. We must have x and we must prevent the denominator from being zero. The denominator factors as x 6x + 8 = (x )(x 4), so x cannot be or 4. So our answer is all real numbers at least as big as and not equal to or 4. In interval notation, our answer is: The domain is [, ) (, 4) (4,.) (END) 7 / 7