Inverse Relations Objectives: Students will be able to… Determine whether a function has an inverse Write an inverse function Verify 2 functions are inverses of each other Inverse Relations

The inverse of a given function will “undo” what the original function did. For example, let’s take a look at the square function: f(x) = x2 x y f-1(x) f(x) 9 3 3 9 9 3 3 9 9 3 3 9 9 3 3 x2 9 9 x 3 3 9 9 9 3 3 3 9 9

Inverse Relations: The inverse of f is f -1 (read “f inverse”) If both the original relation and the inverse relation are both functions, they are inverse functions! The domain of the original relation is the range of the inverse. Inverse Relations:

Graphically, the x and y values of a function and its inverse 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 x 1 2 4 8 16 y 3 Where is there a line of reflection?

y = f(x) y = x The graph of a function and its inverse are mirror images about the line y = f-1(x) y = x

Find the inverse relation: x -3 -1 5 7 10 y -5 4 x y Find the inverse relation:

Writing the equation of an inverse function: Example 1: y = 6x - 12 Step 1: Switch x and y: x = 6y - 12 Step 2: Solve for y:

Find an equation for the inverse relation: 1. 2. Find an equation for the inverse relation:

Verifying 2 functions are inverses If f and g are inverse functions, their composition would simply give x back. Verifying 2 functions are inverses

Ex: Verify that f(x)=-3x+6 and g(x)=-1/3x+2 are inverses. Meaning find f(g(x)) and g(f(x)). If they both equal x, then they are inverses. Because f(g(x))=x and g(f(x))=x, they are inverses. f(g(x))= -3(-1/3x+2)+6 = x-6+6 = x g(f(x))= -1/3(-3x+6)+2 = x-2+2 = x

Verify that the functions f and g are inverses of each other. Since both of these = x, if you start with x and apply the functions they “undo” each other and are inverses.

Remember… The graph of a function needs to pass the vertical line test in order to be a function.

Used to determine whether a function’s inverse will be a function by seeing if the original function passes the horizontal line test. If the original function passes the horizontal line test, then its inverse will pass the vertical line test and therefore is a function. If the original function does not pass the horizontal line test, then its inverse is not a function (may need to restrict domain so that it has an inverse). Horizontal Line Test

All functions have only one y value for any given x value (this means they pass the vertical line test) One-to-one functions also have only one x- value for any given y-value. One to One Functions

If a horizontal line intersects the graph of an equation more than one time, the equation graphed is NOT a one-to-one function and will NOT have an inverse function. This is a one-to-one function This is NOT a one-to-one function This is NOT a one-to-one function

Graph does not pass the horizontal line test, therefore the inverse is not a function. Ex: Graph the function f(x)=x2 and determine whether its inverse is a function.

But what if I restricted the domain of x2 But what if I restricted the domain of x2? Then it would have an inverse. Graph y = x2 for x > 0. Find the inverse function.

Ex: Graph f(x)=2x2-4. Determine whether f -1(x) is a function. How could I restrict the domain to make it have an inverse function? x > 0 f -1(x) is not a function.

Example 2: Given the function : y = 2x2 -4, x > 0 find the inverse Step 1: Switch x and y: x = 2y2 -4 Step 2: Solve for y: Only need the positive root for inverse!

Ex: Write the inverse of g(x)=2x3 y=2x3 x=2y3 Inverse is a function!

Graph the functions to determine whether their inverses will also be functions 𝑦= 𝑥 2 +8 𝑦=± 𝑥 2 −5 𝑦=− 𝑥 4 − 𝑥 2 +4𝑥−3 𝑦= 𝑥 2 +4𝑥−2, 𝑥≥−2 𝑦= 𝑥 3 +7 𝑥 2 −10 𝑦= sin 𝑥