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Functions and Inverse Functions Definition of a function is a set of order pairs for which there is exactly one value of y for each value of x, given that the order pair is ( x, y ) . For example, a set of points such as {(1,3) , (2,4 )} would be considered a function, that is for each value of x would be assigned a unique value of y. This next set would not be a function {(1,3) , (1,4)} , notice that one value of x has two different y values. A better way to see this is plotting the points a making a graph; the Vertical Line Test can help identify which graphs are functions. Below are sets of points to illustrate the definition of a function. Suppose you have sets of ordered pairs such as: Set A A = {(− 3,−2 ) , (- 1,1) , (1,1) , (2,1) , (3,4 )} Draw a straight vertical line through the points. The graph of set A does not hit twice on any vertical line drawn through the graph. This is a Function. B = {(1,1) , (1,3) , (1,-2) , (1,4 ) , (3,4 )} Draw a straight vertical line at (1,1) . The graph of set B is hitting more than one point at x=1. This is Not a Function. C = {(1,1) , (- 1,2 ) , (- 2,-2) , (1,1) , (3,4 )} Although (1,1) repeats in set C, it’s still a function. There are no repeating y values for one single x. This is a Function. The Math Center ■ Valle Verde ■ Set B Set C Tutorial Support Services ■ EPCC 1 The following are examples to practice and helpful to understand what defines a function. The Vertical Line test is used to check if these are functions. Not a Function Function Not a Function Function Function Not a Function Function The Math Center ■ Valle Verde ■ Tutorial Support Services ■ EPCC 2 Inverse of a Function If f (x ) and g ( x ) are functions such that f [g ( x )] = x and g [ f ( x )] = x , then g ( x ) is the inverse of f ( x ) and g ( x ) is notated as f −1 ( x ) . Example 1: Suppose f ( x ) = x − 2 and g ( x ) = x 2 + 2 Check to see if g ( x ) is an inverse function of f (x ) . 2nd: find g [ f ( x )] = x 1st: find f [g ( x )] = x g (x ) = x 2 + 2 f (x ) = x − 2 Substitute x 2 + 2 in for g ( x ) f [g ( x )] = f x 2 + 2 [ [ ] (x f x +2 = f [g ( x )] = x 2 ] 2 ) +2 −2 = x Substitute x − 2 in for f (x ) [ x − 2] g[ x − 2 ] = ( x − 2 ) + 2 = x − 2 + 2 g [ f ( x )] = g 2 2 g [ f ( x )] = x f [g ( x )] = x and g [ f ( x )] = x , by definition g ( x ) is the inverse function of f ( x ) , so g ( x ) = f −1 ( x ) . In order to find an inverse of a function; graphically substitute the x values for the y values. Simply change all the x variables into y’s and all the y variables into x’s. The following is an example: the points flip on the y=x, in other words, the points reflect over the line y=x. The inverses are displayed with squares in Graph A. A = {(− 3,−2) , (− 1,1) , (1,1) , (2,1) , (3,4)} Graph A Set A x -3 -1 1 2 3 y -2 1 1 1 4 A − 1 = {(− 2,−3) , (1,−1) , (1,1) , (1,2 ) , (4,3)} Set A −1 1x 4 -2 1 1 2y 3 -3 -1 1 The Math Center ■ Valle Verde ■ Tutorial Support Services ■ EPCC 3 The following are two examples to find inverses of a function. x−2 4 Substitute all x variables for y’s and y variables for x’s. y−2 x−2 h( x ) = y = ⇒x= 4 4 Find the inverse of h( x ) = Find the inverse of f ( x ) = 2 x 3 + 2 Substitute all x variables for y’s and y variables for x’s. f (x ) = y = 2 x 3 + 2 ⇒ x = 2 y 3 + 2 Then Solve for y. (4)x = ⎛⎜ y − 2 ⎞⎟ (4) ⎝ 4 ⎠ 4x = y − 2 4x + 2 = y − 2 + 2 4x + 2 = y Then solve for y. x = 2 y3 + 2 x − 2 = 2 y3 + 2 − 2 x − 2 = 2y3 x − 2 2y3 = 2 2 x−2 = y3 2 Take the cube root of both sides of equation. x−2 3 3 3 = y 2 3 x−2 =y 2 From this example, the inverse of the function is h −1 ( x ) = 4 x + 2 . The graph below illustrates. ⇒ The inverse of the function is f −1 ( x ) = 3 x−2 2 The graph below illustrates. f (x ) y=x y=x h −1 ( x ) f −1 The Math Center (x ) ■ h( x ) Valle Verde ■ Tutorial Support Services ■ EPCC 4