Inverse Functions Inverses Inverses Now think about functions

Inverse Functions

Inverses

Inverses Now think about functions, like f(x) = x + 1 What function would undo f(x)? What does this mean? Let’s look at a table of values. x f(x) 4 5 -8 -7 0 1 213 214 It would make sense that the inverse function to f(x) should take f(x)’s output back to its input. x f(x)’s inverse 4 5 4 -8 -7 -8 0 1 0 213 214 213 Thus, f(x)’s inverse must be g(x) = x – 1 What happens if we compose these two functions (f ∘ g)(x)? f(g(x)) = g(x) + 1 g(x)’s output takes us =x– 1+1 back to f(x)’s input =x

The function f is a set of ordered pairs, (x, y), then the changes produced by f can be “undone” by reversing components of all the ordered pairs. The resulting relation (y, x), may or may not be a function. Inverse functions have a special “undoing” relationship.

Relations, Functions & 1: 1 Functions are a subset of Functions. They are special functions where for every x, there is one y, and for every y, there is one x. Functions Relations 1: 1 Functions Inverse Functions are 1: 1 Reminder: The definition of function is, for every x there is only one y.

x 1200 1300 1400 f(x) 900 1000 1100 x 900 1000 1100 g(x) 1200 1300 1400

Example

Practice (in your INB)

Solution

Finding the Inverse of a Function

Example

Practice (in your INB) Find the inverse of f(x)=7 x-1

Solution

Practice (in your INB)

Example

Practice (in your INB)

Solution

The Horizontal Line Test And One-to-One Functions

Horizontal Line Test b and c are not one-to-one functions because they don’t pass the horizontal line test.

Example Graph the following function and tell whether it has an inverse function or not. Yes, the graph passes the horizontal line test (the function is 1: 1)

Practice (in your INB) Graph the following function and tell whether it has an inverse function or not.

Solution Graph the following function and tell whether it has an inverse function or not. No, the graph does not pass the horizontal line test so the function is not 1: 1 and therefore does not have an inverse (unless we restrict the domain of x)

Graphs of f and f-1

There is a relationship between the graph of a one-to-one function, f, and its inverse f -1. Because inverse functions have ordered pairs with the coordinates interchanged, if the point (a, b) is on the graph of f then the point (b, a) is on the graph of f -1. The points (a, b) and (b, a) are symmetric with respect to the line y=x. Thus graph of f -1 is a reflection of the graph of f about the line y=x.

A function and it’s inverse graphed on the same axis.

Example If this function has an inverse function, then graph it’s inverse on the same graph. f(x) f-1(x) They are the same functions – when you reflect f(x) over the line y = x, it’s the same graph

Practice Example (in your INB) If this function has an inverse function, then graph it’s inverse on the same graph.

Solution If this function has an inverse function, then graph it’s inverse on the same graph. Reflect the graph over the line y = x

Practice Example (in your INB) If this function has an inverse function, then graph it’s inverse on the same graph.

Solution If this function has an inverse function, then graph it’s inverse on the same graph.

Applications of Inverse Functions The function given by f(x)=5/9 x+32 converts x degrees Celsius to an equivalent temperature in degrees Fahrenheit. a. Is f a one-to-one function? Why or why not? F=f(x)=5/9 x+32 is 1 to 1 because it is a linear function. b. Find a formula for f -1 and interpret what it calculates. The Celsius formula converts x degrees Fahrenheit into Celsius. Replace the f(x) with y Solve for y, subtract 32 Multiply by 9/5 on both sides

Practice Problems

(a) (b) (c) (d) Solution next slide

(a) (b) (c) (d) Correct answer is )c(

(a) (b) (c) (d) Solution next slide

(a) (b) (c) (d) Correct answer is )a(