Inverse Functions

Find inverse functions algebraically and graphically. Math 30-1 Alberta Grade 12.

Lesson 1.9 of Function Transformations & Operations in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Understanding Inverse Functions. Learn what inverse functions are, how to find them from points and equations, and understand their graphical properties.
• Working with Inverse Functions. Apply inverse functions to transformations, find parameters using inverse relationships, and master key properties.

Inside the lesson — a free preview

What is an Inverse Function?

Inverse functions are functions that UNDO or REVERSE each other.

Notation.

Important. The -1 is NOT an exponent. It represents "inverse function," not (1)/(f(x)).

How to Find an Inverse

To determine the inverse of a function, we interchange the x- and y-coordinates of a point, and switch the x and y variables in an equation.

Steps.

Graphical Property

The inverse of a relation can be found by reflecting the original relation over the line y = x.

Key Facts.

Domain and Range Swap

If f is a function with domain A and range B, the inverse has domain B and range A.

Original function:.

Inverse function:.

The domain and range switch places.

Horizontal Line Test

The horizontal line test determines if the inverse will be a function.

The Test. If it is possible for a horizontal line to intersect a graph more than once, then the inverse is NOT a function.

Restricting the Domain

If the inverse is not a function, we can restrict the domain of the original function so that the inverse becomes a function.

Example. For f(x) = x^2, the inverse is not a function because it fails the horizontal line test.

But if we restrict the domain to x ≥ 0, then the inverse is a function.

Finding Inverse from Transformations

When working with transformed inverse functions, we need to apply the inverse first (swap coordinates), then apply the transformations to the new point.

Example. If the point P(-2, 5) is on the graph of y = f(x), what point is on y = f^(-1)(x - 3) + 1?

Step 1: Apply the inverse to swap coordinates

(-2, 5) → (5, -2)

Since f(-2) = 5, we know that f^(-1)(5) = -2, so the point (5, -2) is on y = f^(-1)(x)

Step 2: Identify the transformations on the inverse

Step 3: Apply transformations to the point

Start with (5, -2) from f^(-1)(x):

• Horizontal shift right 3: x-coordinate becomes 5 + 3 = 8

• Vertical shift up 1: y-coordinate becomes -2 + 1 = -1

Answer: (8, -1)

Worked example: From Points

Problem. Given f(x) = \(2,3), (3,5), (4,7), (5,9)\, write the inverse.

Swap the coordinates:

Each (x, y) becomes (y, x).

f^(-1)(x) = \(3,2), (5,3), (7,4), (9,5)\

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