Standards: MM2A5 – Students will explore inverses of functions. a. Discuss the characteristics of functions and their inverses, including one-to-oneness, domain, and range. c. Explore the graphs of functions and their inverses.

Functions and Their Inverses EQ: What are the characteristics of functions and their inverses?

Relation: Relation – a set of ordered pairs (or graph). EX: { (0, 1) (-5, 3) ( ½ , 23) (.4, π) } or Domain – x-values Range – y-values

Functions: Function – a relation where every x-value is paired with exactly one y-value. (No x-values can repeat) Vertical-line-test – If a vertical line intersects the relation's graph in more than one place, then the relation is NOT a function. No! Yes!

Inverse of a Relation: Inverse of a Relation – when a relation is taken and all the x-values and y-values have been switched. The x and y coordinates have been switched. EX: relation: {(4, 10) (8, -2) (3, 5) (18, ½ )} inverse of the relation: {(10,4) (-2,8) (5,3) ( ½,18)}

Graphically, the x and y values of a point are switched. If the function y = g(x) contains the points x 1 2 3 4 y 8 16 then its inverse, y = g-1(x), contains the points Y = x x 1 2 4 8 16 y 3 Where is there a line of reflection?

Inverse Function: Inverse Function – The x and y-values have been switched and the resulting relation is a function. Both the function and the inverse function are graphed and are symmetric over the y = x line. Inverse Function Notation - f -1(x) (looks like f is raised to negative one, but is inverse notation)

One-to-One: When a relation is a function and its inverse is a function, then the original function is said to be one-to-one. This means …every x-value is paired with exactly one y-value AND every y-value is paired with exactly one x-value. Horizontal-line-test - If a horizontal line intersects the function’s graph in more than one place, then the function is NOT one-to-one. (which means its inverse is not a function)

Testing a Graph to see if it is One-to-One: 1. You must do the vertical line test to determine if the relation is a function. 2. Then, if it is a function, you can do the horizontal line test to determine if its inverse is a function. Yes! A function. Yes! One-to-One.

Examples: Yes! A function. Yes! A function. Yes! One-to-One. No! Not One-to-One.

Inverses to Linear Functions

The inverse of a given function will “undo” what the original function did. 1. Look at the function F(x) and go through the order of operations as if you were replacing x with a value. 2. Now look at the inverse function F-1(x) and go through its order of operations. F(x) = 2x + 5 F-1(x) = x – 5 2 1. Multiply by 2. 1. Subtract 5. 2. Add 5. 2. Divide by 2.

Complete the table of values: Now Graph: F(x) = 2x + 5 F-1(x) = x – 5 2 X Y 5 1 7 9 -1 3 -2 5 7 1 9 2 3 -1 1 -2

Important facts about Inverses If f is one-to-one, then f -1 exists. The domain of f is the range of f -1, and the range of f is the domain of f -1. If the point (a,b) is on the graph of f, then the point (b,a) is on the graph of f -1, so the graphs of f and f -1 are reflections of each other across the line y = x.

Steps to Finding the Inverse to a Function: 1. Replace f(x) with y. 2. Switch the x and y. 3. Solve the equation for y. 4. Replace the y with f -1(x).

Examples: Find the inverse to the functions 1. f(x) = 3x – 7 2. g(x) = ½ x + 10 y = 3x – 7 x = 3y – 7 + 7 + 7 x + 7 = 3y 3 3 x + 7 = y 3 f -1(x) = x + 7 y = ½ x + 10 x = ½ y + 10 - 10 - 10 x - 10 = ½ y 2(x – 10) = 2 ( ½ )y 2x – 20 = y g -1(x) = 2x - 20

5.1 Example of Finding f -1(x) Example Find the inverse, if it exists, of Solution Write f (x) = y. Interchange x and y. Solve for y. Replace y with f -1(x).

Homework Sheet

Inverses to Power Functions

Inverses work for more than just linear functions. Let’s take a look at the square function: f(x) = x2 x f(x) y f -1(x) x 9 3 3 9 9 3 3 9 9 3 3 9 9 3 3 x2 9 9 3 3 9 9 9 3 3 3 9 9

Power Functions: When a function is written in the form f(x) = xn (where n > 1) it is considered a power function. Examples of Power Functions: 1. f(x) = x2 2. g(x) = x3 3. h(x) = x4 ***How do we “undo” power functions?*** 1. 2. 3.

Find the inverse to the Power functions: 1. f(x) = x3 + 5 2. g(x) = (x – 1)3 - 2 y = x3 + 5 x = y3 + 5 x – 5 = y3 y = (x – 1)3 – 2 x = (y – 1)3 – 2 x + 2 = (y – 1)3

How do I use composition of functions to verify that functions are inverses of each other? If two functions, f(x) and g(x), are given and you need to verify that they are inverses… You will have to show that the composition of (f o g)(x) = x AND (g o f)(x) = x **This is because the two functions “undo” each other and what you put in for x in the original function will be undone by the inverse. Therefore, you will get x again.

Example: Verify that f(x) = 5x + 1 and g(x) = x – 1 are inverses 5