Inverse FunctionsWinnie Chen, Gary Choi, Kayla Glufling, Jacky Chen

What is an “inverse function”?A function that performs the REVERSE of the original function. Therefore, when the inverse is plugged in as X in the original equation, the answer would be y=x (vice versa).

ƒ(g(x)) = x AND g(ƒ(x)) = x

The function g would be denoted as ƒ-1 and read as “ƒ inverse”.

How to find an inverse functionWrite the original relation y = 2 x — 4 Switch x and y x = 2 y — 4Add 4 to both sides x + 4 = 2 y Dive both sides by 2 ½ x + 2 = y

The inverse relation of y = 2 x – 4 is y = ½ x + 4

With any given function, you can find its inverse by switching the places of x and y, then simply solve for y.

How to verify an inverse functionVerify that ƒ(x) = 2x—4 and ƒ-1 (x) = ½x+2

Using ƒ(ƒ-1(x)) = x Plug in the inverse into ƒ-1 ƒ(ƒ-1(x)) = ƒ(½x+2)Plug in the original ƒ(ƒ-1(x)) = 2(½x+2)—4 Simplify ƒ(ƒ-1(x)) = x + 4 — 4 Solve ƒ(ƒ-1(x)) = x

Using ƒ-1(ƒ(x)) = xPlug in the original into ƒ ƒ-1(ƒ(x)) = ƒ-1(2x—4)Plug in the inverse ƒ-1(ƒ(x)) = ½(2x—4) +2 Simplify ƒ-1(ƒ(x)) = x—2 +2Solve ƒ-1(ƒ(x)) = x

Input/output relation• The DOMAIN of the inverse relation is the RANGE of the

original relation. • The RANGE of the inverse relation is the DOMAIN of the

original relation.

X -2 -1 0 1 2

Y 4 2 0 -2 -4

X 4 2 0 -2 -4

Y -2 -1 0 1 2

So, what does the graph look like?

The graph of the inverse relation is simply the reflection of graph of the original relation. Therefore the line of reflection would be y = x

**You can find the inverse relationby using the graph. Just switchthe range and domain of the original equation.

Original

Line of Symmetry

Inverse

How to find inverse of power functions

Write the original relation: f(x)= 1/16x5

Switch x and y: x= 1/16y5

Multiply both sides by 16: 16*x = y5

Take both sides to the 1/5 power: (16x)1/5 = (y5)1/5

Simplify: (16x)1/5 = y

Solve: y = 0.2x1/5

How to find the inverse of a cubic function • Write the original function: f(x) = x3+4

• Substitute y into f(x): y = x3+4

• Switch x and y: x = y3+4

• Minus 4 on both sides: x – 4 = y3

• Cube root both sides: 3√(x-4) = y

• Substitute f-1(x) for y: f-1(x) = 3√(x-4)