INVERSE FUNCTIONS

After learning this topic you will be able… to recognize from the graph of a function whether the function has an inverse; to recognize from the graph of a function whether the function is one to one; to graph the inverse of a function; to algebraically find the inverse of a function;

Relation vs Function A “relation” is just a relationship between sets of data or information. A “function” is a well-behaved relation. This means that all functions are relations & not all relations are functions

To Function or Not to Function

To Function or Not to Function

To Function or Not to Function

To Function or Not to Function

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

function function Not a Function x1 y1 x1 y1 x2 y2 x2 x3 x3 y3 y3 Domain Range Domain Range One-to-one function NOT One-to-one function x1 y1 Not a Function Any one “x” can’t have more than 1 “y” y2 x3 y3 Domain Range

To determine by the graph if an equation is a function, we have the vertical line test. If a vertical line intersects the graph of an equation more than one time, the equation graphed is NOT a function. This is NOT a function This is a function This is a function

Horizontal Line Test to see if the inverse is a function. If the inverse is a function, each y value could only be paired with one x. Let’s look at a couple of graphs. So this does not have an inverse function! We can graph the inverse, but it is not a function. This does have an inverse function. Horizontal Line Test to see if the inverse is a function.

This means “inverse function” Graph: 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

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

Given the function : y = 3x2 + 2 find the inverse: Example 2: Given the function : y = 3x2 + 2 find the inverse: Step 1: Switch x and y: x = 3y2 + 2 Step 2: Solve for y:

Ex: Find an inverse of y = -3x+6. Steps: -switch x & y -solve for y y = -3x+6 x = -3y+6 x-6 = -3y

Graph f(x) = 3x − 2 and using the same set of axes. Then compare the two graphs. Determine the domain and range of the function and its inverse.

Consider the graph of the function The inverse function is

Consider the graph of the function The inverse function is x x x x The inverse function is An inverse function is just a rearrangement with x and y swapped. So the graphs just swap x and y!

What else do you notice about the graphs? x x is a reflection of in the line y = x The function and its inverse must reflect on y = x