Intermediate Algebra Section 9. Composite Functions and Inverse Functions We have added, subtracted, multiplied, and divided functions in previous chapters. Another way to combine functions is called composite functions. To understand this, consider the tables to the right. The first table shows x =Degrees Fahrenheit (Input) C x =Degrees ( ) Celsius (Output) degrees Celsius C ( x ) as a function of degrees Fahrenheit x. 3 3 68 49 5 0 0 65 The second table shows Kelvins K C as a ( ) C =Degrees Celsius (Input) K C =Kelvins ( ) (Output) function of degrees Celsius C. 5 0 0 65 48.5 73.5 93.5 385.5 The third table takes us directly from Fahrenheit to Kelvins. x =Degrees Fahrenheit(Input) K C x =Kelvins ( ( )) (Output) 3 3 68 49 48.5 73.5 93.5 385.5 Notice the output of the first table is the is the same as the input of the second table.
Section 9. Composite Functions and Inverse Functions page Composition of Functions ( ) The composition of functions f and g is ( f g )( x) = ( ) domain of the composite function is the set of all x in the domain of g such that g ( x ) is in the domain of f. f g x. The Find the domain of the composition of f with g when f ( x) = x and g ( x) = x. ( )( ) = ( ( )) f g x f g x Definition of f g ( ) ( ) ( ) = f x g x = x is the inner function. = x Input x into the outer function f. = x, x 0 Domain of f g is all x 0 Caution: Multiplication of functions and composition of functions are two completely different operations. Be sure to recognize the difference between the multiplication and the composition of two functions. Example: If f ( x) = x 6x + and h( x) = x, find ( f h)( ).
Section 9. Composite Functions and Inverse Functions page 3 Example: If f ( x) = x 3 and g ( x) = x find ( f g )( x) ( g f )( x). and Example: If f ( x) = x + 0 and g ( x) = 3x + find ( f g )( x) ( g g )( x). and
Section 9. Composite Functions and Inverse Functions page 4 A special classification of functions are one-to-one (-) functions. Definition of One-To-One Functions For a one-to-one function, each x -value (input) corresponds to only one y -value (output), and each y -value (output) corresponds to only one x -value (input). That is, if x and x are two different inputs of a function f, then f x f x. ( ) ( ) Example : Determine whether the following functions are one-toone functions. a) Student Number of courses currently taking { } b) r = (, ), ( 3,4 ), ( 5,6 ), ( 6,7)
Section 9. Composite Functions and Inverse Functions page 5 { } c) g = ( 8,6 ), ( 9,6 ), ( 3,4 ), ( 4,4) know that if an equation passes the vertical line test then the equation is a function f. It turns out that, if a function f is one to one, then the function f must pass the horizontal line test. Horizontal Line Test for Inverse Functions A function f has an inverse function f if and only if f is one-to- one. Graphically, a function f has an inverse function f if and only if no horizontal line intersects the graph of f at more than one point.
Section 9. Composite Functions and Inverse Functions page 6 Example: Determine whether each graph is the graph of a one-toone function. a) b) c) d)
Section 9. Composite Functions and Inverse Functions page 7 For any one-to-one function, we can find its inverse function by switching the coordinates of the ordered pairs of the functions, or the inputs and the outputs. Definition of Inverse Function Let f and g be two functions such that ( ( )) ( ( )) f g x = x for every x in the domain of g and g f x = x for every x in the domain of f. The function g is called the inverse function of the function f, and is denoted by ( ( )) f f x x f, and vice versa. ( ) f (read f -inverse ). So, ( ) f f x = x and =. The domain of f must be equal to the range of Example: If f ( x) = 3x 0, show that f ( x) x 0 =. 3 +
Section 9. Composite Functions and Inverse Functions page 8 Example: For the functions in Example, find the inverse function of those that are one-to-one. If a one-to-one function f is defined as a set of ordered pairs, we can find f by interchanging the x - and y - coordinates of the ordered pairs. If a one-to-one function f is given in the form of an equation, we use the following procedure to find f. Finding the Inverse of a One-To-One Function f ( x ). In the equation for f ( x ), replace ( ) f x with y.. Interchange the roles of x and y. 3. If the new equation does not represent y as a function of x, the function f does not have an inverse function. If the new equation does represent y as a function of x, solve the new equation for y. 4. Replace y with f ( x) 5. Verify that f and f showing that ( ). are inverse functions of each other by ( ) ( ( )) f f x = x = f f x. f x = x Example: Find an equation of the inverse of ( ) 3
Section 9. Composite Functions and Inverse Functions page 9 Example: Find the equation of the inverse of ( ) Graph f ( x ) and its inverse on the same set of axes. f x = x +. y 7 6 5 4 3-7 -6-5 -4-3 - - 3 4 5-6 7 x - -3-4 -5-6 -7
Section 9. Composite Functions and Inverse Functions page 0 Notice that the graphs of f and f are mirror images of each other. In general, if the point ( a, b ) lies on the graph of f, then the point ( b, a ) must lie on the graph of the graph of y = x. f f, and vice versa. This means that is a reflection of the graph of f about the line This means that if we want to graph the inverse of a function f, we can graph f and then just reflect the graph of f about the line y = x. Example: For the functions in Example 3, graph the inverse function of those that are one-to-one. a) b)
Section 9. Composite Functions and Inverse Functions page c) d)