Warm Up 8/17/17 Find the equation of the line with Slope:5 and Point: 3,6

Inverse Functions

Inverse: The opposite. The inverse undoes what the original function does Here are some “opposites:” + − × ÷ 𝑥 2 𝑥 4,8 8,4 𝑓(𝑥) 𝑓 −1 (𝑥)

𝑓(𝑥) denotes the original function 𝑓 −1 (𝑥) denotes the inverse function

For example, let’s take a look at the square function: f(x) = x2 In the same way, 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 f(x) y f-1(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

For example, let’s take a look at the square function: f(x) = x2 In the same way, 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) 5 25 5 5 25 5 25 5 5 25 25 5 5 x2 5 25 25 5 5 25 25 5 25 5 5 5

For example, let’s take a look at the square function: f(x) = x2 In the same way, 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 f-1(x) f(x) y 11 121 11 11 121 121 11 11 121 121 11 11 x2 121 121 11 121 11 121 11 121 11 11 121 121 121 121 11 11

Graphically, the x and y values of a point are switched. The point (4, 7) has an inverse point of (7, 4) AND The point (-5, 3) has an inverse point of (3, -5)

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

Graph Linear Equations (and their inverse) 𝑦=3𝑥−6 3 𝑜𝑟 3 1 Slope: y-int: −6 Inverse: 𝑥=3𝑦−6 +6 +6 𝑥+6=3𝑦 𝑥 3 + 6 3 = 3𝑦 3 1 3 𝑥+2=𝑦

Find the Inverse A. 𝑦=4𝑥−2 B. 𝑦= 𝑥 5 +8 𝑥= 𝑦 5 +8 𝑥=4𝑦−2 +2 +2 −8 −8 +2 +2 −8 −8 𝑥+2=4𝑦 𝑥−8= 𝑦 5 𝑥 4 + 2 4 = 4𝑦 4 5(𝑥−8)= 𝑦 5 ∗5 𝑥 4 + 1 2 =𝑦 5𝑥−40=𝑦

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:

Example 2: Given the function : find the inverse: Step 1: Switch x and y: Step 2: Solve for y: