5.2 Inverse Functions and Their Representations Calculate inverse operations Identify one-to-one functions Find inverse functions symbolically Use other representations to find inverse functions

Example: Finding inverse actions and operations For each of the following, state the inverse actions or operations. Put on a coat and go outside. Subtract 5 from x and divide the result by 2. Solution Reverse order, apply inverse actions Come inside and take off coat. Multiply x by 2 and add 5.

Inverse Functions (1 of 2) Gallons to Pints f(x) = 8x x 1 2 3 4 f(x) 8 16 24 32 x 8 16 24 32 f −1(x) 1 2 3 4 Function g performs the inverse operation of f.

Inverse Functions (2 of 2) Inputs and outputs (domains and ranges) are interchanged for inverse functions.

Composition with Inverse Functions The composition of a function with its inverse using input x produces output x.

Determine If a Function Has an Inverse If different inputs of a function f produce the same output, then an inverse function of f does not exist. However, if different inputs always produce different outputs, f is a one- to-one function. Every one-to-one function has an inverse function.

One-to-One Function A function f is a one-to-one function if, for elements c and d in the domain of f, c ≠ d implies f(c) ≠ f(d). That is, different inputs always result in different outputs.

Example: Determining if a function is one-to-one graphically (1 of 3) Use each graph to determine if f is one-to-one and if f has an inverse function.

Example: Determining if a function is one-to-one graphically (2 of 3) Solution a. The horizontal line y = 2 intersects the graph of f at (−1, 2), (1, 2) and (3, 2). This means that f(−1) = f(1) = f(3) = 2. Three distinct inputs, −1, 1, and 3, produce the same output, 2. Therefore f is not one-to-one and does not have an inverse function.

Example: Determining if a function is one-to-one graphically (3 of 3) Solution b. Because every horizontal line intersects the graph at most once, different inputs (x-values) always result in different outputs (y-values). Therefore f is one-to- one and has an inverse function.

Horizontal Line Test If every horizontal line intersects the graph of a function f at most once, then f is a one-to-one function.

Increasing, Decreasing and One-to-One Functions If a continuous function f is always increasing on its domain, then every horizontal line will intersect the graph of f at most once. By the horizontal line test, f is a one-to-one function. Similarly, if a continuous function g is only decreasing on its domain, then g is a one-to-one function.

Inverse Function for every x in the domain of f and

Example: Finding and verifying an inverse function (1 of 5) Let f be a one-to-one function given by f(x) = x³ − 2.

Example: Finding and verifying an inverse function (2 of 5) Let f(x) = x³ − 2. Solution a. Solve y = f(x) for x.

Example: Finding and verifying an inverse function (3 of 5) Let f(x) = x³ − 2. Both the domain and the range of the cube root function include all real numbers. The graph of is the graph of the cube root function shifted left 2 units.

Example: Finding and verifying an inverse function (4 of 5)

Example: Finding and verifying an inverse function (5 of 5) and now

Finding a Symbolic Representation for f −1

Numerical Representation of Inverse (1 of 3) In the top table, f computes the percentage of the U.S. population with 4 or more years of college in year x. x 1940 1970 2015 f(x) 5 11 34 x 5 11 34 f-1(x) 1940 1970 2015

Numerical Representation of Inverse (2 of 3) Function f has an inverse function.

Numerical Representation of Inverse (3 of 3) The domains and ranges are interchanged. This relation does not represent the inverse function because input 4 produces two outputs, 1 and 2.

Domains and Ranges of Inverse Functions

Graphs of Functions and Their Inverses

Example: Representing an inverse function graphically (1 of 2) Solution Calculator display. [−5,5,1] by [−5,5,1]

Example: Representing an inverse function graphically (2 of 2)