From Ordered Pairs

If a relation is given as a set of ordered pairs, its inverse is found by swapping each pair — changing (x, y) to (y, x).

Example: Given R = {(2, 4), (3, 6), (5, 10)}, find R⁻¹.

Solution: Swap each ordered pair: R⁻¹ = {(4, 2), (6, 3), (10, 5)}

From an Equation

If a relation is given by an equation involving x and y, find its inverse by swapping x and y, then rewrite the equation with y as the subject.

Example 1: Find the inverse of R = {(x, y) : y = 2x + 1}

Solution:

1. Swap x and y: x = 2y + 1
2. Solve for y: x − 1 = 2y y = (x − 1)/2

So R⁻¹ = {(x, y) : y = (x − 1)/2}

Example 2: Find the inverse of R = {(x, y) : x + 5 ≥ y}

Solution:

1. Swap x and y: y + 5 ≥ x
2. Rewrite: x ≤ y + 5, which is the same as y ≥ x − 5

So R⁻¹ = {(x, y) : y ≥ x − 5}

Domain and Range of Inverse Relations

When you invert a relation, the domain and range swap:

• Domain of R = Range of R⁻¹
• Range of R = Domain of R⁻¹

A function is a special type of relation where each x-value produces exactly one y-value. Not every function has an inverse that is also a function. This only happens when the function is one-to-one.

One-to-One Functions and the Horizontal Line Test

A function is one-to-one if no horizontal line intersects its graph more than once. This test determines whether the inverse is also a function.

If the horizontal line test passes — the function has an inverse that is also a function.

If the horizontal line test fails — the inverse is a relation but not a function.

Finding the Inverse of a Function Algebraically

When a function is one-to-one, its inverse f⁻¹ is found by:

1. Write the function as y = f(x)
2. Swap x and y
3. Solve for y (make y the subject)

Example 1: Find the inverse of f(x) = 3x − 6

Solution:

1. Write as y = 3x − 6
2. Swap x and y: x = 3y − 6
3. Solve for y: x + 6 = 3y y = (x + 6)/3

Therefore, f⁻¹(x) = (x + 6)/3

Example 2: Find the inverse of f(x) = x³

Solution:

1. Write as y = x³
2. Swap x and y: x = y³
3. Solve for y: y = ∛x

Therefore, f⁻¹(x) = ∛x

Domain and Range of Inverse Functions

For any function f and its inverse f⁻¹:

• Domain of f⁻¹ = Range of f
• Range of f⁻¹ = Domain of f

This makes sense because the inverse "undoes" what the original function did — it takes the output back to the original input.

Graphical Representation

The graph of f⁻¹ is the reflection of the graph of f across the line y = x. This is because swapping x and y in the coordinate pairs has the geometric effect of reflecting over the line y = x.

Problem: Find the inverse of f(x) = x − 5, then state its domain and range.

Solution:

1. Write as y = x − 5
2. Swap x and y: x = y − 5
3. Solve for y: y = x + 5

Therefore, f⁻¹(x) = x + 5

Domain: Since f is a linear function with domain all real numbers, f⁻¹ also has domain all real numbers.

Range: Similarly, the range of f⁻¹ is all real numbers.

Verification: f(2) = 2 − 5 = −3, and f⁻¹(−3) = −3 + 5 = 2. The inverse returns the original input.

In everyday life, inverse functions are used when converting between different measurement systems. For example, if a market vendor in Dar es Salaam sets prices in Tanzanian shillings based on a formula like price = 500 × quantity, a buyer can use the inverse function (quantity = price ÷ 500) to determine how many items they can afford for a given amount of money. This reversing of a relationship helps in planning purchases and managing budgets.