Warm-Up For the following, make a T-Chart and sketch a graph for x ={-2, -1, 0, 1, 2}

Finding Inverses by Graphing

Things to know… The inverse of a function is denoted by To graph an inverse: Make a table, switch x and y, re-graph The inverse is the graph reflected over the line y=x. THIS MEANS THAT THE DOMAIN of f(x) becomes the range of the inverse and the RANGE of f(x) becomes the domain of the inverse.

THIS INVERSE IS NOT A FUNCTION THIS INVERSE IS NOT A FUNCTION! How can we look at the original to determine if its inverse will be a function? x y 4 -2 1 -1 2 x y -2 4 -1 1 2

x y 1 2 4 3 9 16 x y 1 4 2 9 3 16

INVERSE FUNCTIONS

Remember we talked about functions---taking a set X and mapping into a Set Y 1 2 3 4 5 10 8 6 1 2 2 4 3 6 4 8 10 5 Set X Set Y An inverse function would reverse that process and map from SetY back into Set X

If we map what we get out of the function back, we won’t always have a function going back!!! 1 2 2 4 3 6 4 8 5 A relation is called a function when each input (x) maps to only one output (y). A one-to-one function is a special type of function where each input maps to a unique output (no repeats!)

Relations and Functions Relation – a mapping of input values (x-values) onto output values (y-values). Here are 3 ways to show the same relation. x y -2 4 -1 1 0 0 1 1 y = x2 Equation Table of values Graph

Inverse relation – just think: switch the x & y-values. -2 -1 0 0 1 1 x = y2 ** the inverse of an equation: switch the x & y and solve for y. ** the inverse of a table: switch the x & y. ** the inverse of a graph: the reflection of the original graph in the line y = x.

Graph f(x) and f -1(x) on the same graph. -4 5 5 -4 3 3 4 5 5 4

Graph f(x) and f -1(x) on the same graph. -4 -4 -4 -4 -3 -5 -5 -3 4 4

Let’s consider the function and compute some values and graph them. Notice that the x and y values traded places for the function and its inverse. These functions are reflections of each other about the line y = x Let’s consider the function and compute some values and graph them. This means “inverse function” x f (x) (2,8) -2 -8 -1 -1 0 0 1 1 2 8 (8,2) x f -1(x) -8 -2 -1 -1 0 0 1 1 8 2 Let’s take the values we got out of the function and put them into the inverse function and plot them (-8,-2) (-2,-8) Is this a function? Yes What will “undo” a cube? A cube root