7.4 Inverses of Functions

+ − − + × ÷ ÷ × x2 x2 x3 x3 We know what inverse operations are: We use inverse operations to solve equations when we need to cancel something. × ÷ ÷ × x2 Functions may have an inverse also. x2 is read “f inverse of x”. x3 Don’t confuse this with a negative exponent. x3

Example 1. Find the inverse of f(x) = 2x + 2. Steps 1. Replace f(x) with y. y y = 2x + 2 x 2. Switch x and y. x = 2y + 2 −2 −2 3. Solve for y. x – 2 = 2y 2y = x – 2 2 2 4. Replace y with

Now, let’s take a look at the graphs of f(x) = 2x + 2 and (1, 4) f(x) = 2x + 2 (0, 2) m = 2 b = 2 (−1, 0) (4, 1) (2, 0) (0, −1) m = b = −1 Now, take a look at the coordinates. What do you notice about the coordinates?

What do you notice about the graphs? Using a graphing calculator, graph the pairs of equations on the same graph. Sketch your results. Be sure to use the negative sign, not the subtraction key. These graphs are said to be inverses of each other. What do you notice about the graphs?

An inverse relation “undoes” the relation and switches the x and y coordinates. In other words, if the relation has coordinates (a, b), the inverse has coordinates of (b,a) Inverse of Function f(x) Function f(x) X Y 3 1 4 -3 -5 2 5 -8 X Y 3 4 1 -3 2 -5 5 -8

Let’s look at our graphs from earlier Let’s look at our graphs from earlier. Notice that the points of the graphs are reflected across a specific line. What is the equation of the line of reflection? y = x

You can tell if two functions, such as f(x) and g(x) are inverses only if Example 2. Determine whether f(x) = −3x + 6 and are inverses. Since f(x) and g(x) are inverses.

Find the inverse of a function : Example 3: y = 6x - 12 Step 1: Switch x and y: x = 6y - 12 Step 2: Solve for y:

Example 4: Given the function : y = 3x2 + 2 find the inverse: Step 1: Switch x and y: x = 3y2 + 2 Step 2: Solve for y:

Do all functions have inverses? Yes, and no. Yes, they all will have inverses, BUT we are only interested in the inverses if they are FUNCTIONS. DO ALL FUNCTIONS HAVE INVERSES THAT ARE FUNCTIONS? Recall, functions must pass the vertical line test when graphed. If the inverse is to pass the vertical line test, the original function must pass the __________ test!

Horizontal Line Test Not all functions have inverses. You can tell by graphing the function and drawing a horizontal line. If the horizontal line intersects more than one point, then the function does not have an inverse. Passing the horizontal line test (and the vertical line test), means the original function and it’s inverse are “one-to-one”. That means there’s exactly one y for every x and pne x for every y.

Example 5. Graph the function f(x) = 2x3 – 4. Find the inverse of f(x) if it exists. x y −4 1 −2 −1 −6 2 12 −2 −20 Draw a horizontal line to see if f(x) has an inverse. Since f(x) passes the horizontal line test, find the inverse.

Example 5. Graph the function f(x) = 2x3 – 4. Find the inverse of f(x) if it exists. f(x) = 2x3 – 4 y = 2x3 – 4 x = 2y3 – 4 2y3 – 4 = x 2y3 = x + 4