Precalculus 1.7 INVERSE FUNCTIONS

DO THIS NOW! You have a function described by the equation: f(x) = x + 4 The domain of the function is: {0, 2, 5, 10} YOUR TASK: write the set of ordered pairs that would represent this function **use the equation to find f(x) for each number in the domain Graph the points in the function.

Functions as Sets of Ordered Pairs Recall that besides describing functions with equations or graphs, we can do so by listing the ordered pairs that make up the function

Inverse function Notation: f-1 is the inverse of f Definition: A function’s inverse brings output values back to their input values. f(x)=x+4 6 2 INPUT OUTPUT f-1(x)=x-4

Inverse Functions: 3 representations GRAPH ALGEBRA ORDERED PAIRS

Inverse Functions: Ordered Pairs Original function, f: {(0, 4) (2, 6) (5, 9) (10, 14)} Inverse function, f-1: {(4, 0) (6, 2) (9, 5) (14, 10)} What is f(2)? What is f-1(6)? To find the inverse function, represented by ordered pairs, simply flip each ordered pair If f contains (x, y), then f-1 contains (y, x).

Inverse Functions: Algebra The equation of the inverse function should “undo” the equation of the original function. Ex: If f(x) = x + 4, then f-1(x) = x – 4 Ex: If g(x) = 4x, then g-1(x) = …?

Precise Definition 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. Then g is the inverse of the function f. g = f-1 The domain of f is the range of f-1. The range of f is the domain of f-1.

A logical point: If f is the inverse of g, then g is the inverse of f. Furthermore, g and f are inverses of each other.

Example Which of these is the inverse of:

Inverse Function: GRAPHS Graph these functions and their inverses on the same graph. f(x) = x + 4 and g(x) = x – 4 f(x) = 4x and g(x) = x/4 f(x) = .5(x+3) and g(x) = 2x – 3 f(x) = x2 and {(0,1) (2,5) (3,6) (4,8)} and {(1,0) (5,2) (6,3) (8,4)}

Inverse Graphs The graph of a f and f-1 are related in a special way. If (x,y) is on f’s graph, (y,x) must be on f-1’s graph. Therefore, the graph of f-1 is a reflection of the graph of f across the line y = x.

Practice What is the inverse of: f(x) = 2x + 4

Which functions have inverses? What is this function’s inverse? (0, 1) (2, 4) (3, 4)

Functions without inverses If multiple input values have the same output value, then the function has no inverse This is because given a repeated output, there would be no way to tell what the original input was f: (0,1) (2,4) (3,4) f-1: (1,0) (4,2) (4,3) This is NOT a function! Therefore we have no surefire way to undo f.

Functions without inverses: a list f(x) = x2 (What was x if f(x)=4?) f(x) = xn where n is even f(x) = |x|

Functions without inverses: graph test If a function, f, has an inverse, f-1, then the inverse is also a function. Therefore f-1 must pass the vertical line test. In order for f-1 to pass the vertical line test, f must pass the HORIZONTAL LINE TEST. f-1 f

Will these functions have inverses? 1) 2) 3) 4) If a function both increases and decreases, can it have an inverse?

Finding the inverse of a function algebraically Use the horizontal line test to decide whether f has an inverse. In the equation, replace f(x) with “y.” Switch “x” and “y.” Solve for y. Replace y with “f-1(x).” Check your work! 1. 2. 3. 4. 5.

Practice: find the inverses of these functions 1) 2) 3) 4) 5) 6)