Which Of The Following Is The Inverse Of

Which of the Following Is the Inverse Of: A Complete Guide to Finding and Identifying Inverse Functions

Understanding inverse functions is one of the most fundamental skills in algebra and precalculus. That said, whether you are preparing for a standardized test, a college placement exam, or simply trying to strengthen your math foundation, knowing how to determine which of the following is the inverse of a given function is an essential competency. This article will walk you through everything you need to know — from the basic concept to step-by-step problem-solving strategies.

What Is an Inverse Function?

An inverse function essentially "undoes" what the original function does. If a function f(x) takes an input x and produces an output y, then the inverse function, written as f⁻¹(x), takes that output y and returns the original input x.

Think of it like this: if a function wraps a gift, the inverse function unwraps it. They are mirror images of each other across the line y = x.

Key Properties of Inverse Functions

Before diving into how to find an inverse, let's review the core properties:

Domain and range swap: The domain of f(x) becomes the range of f⁻¹(x), and the range of f(x) becomes the domain of f⁻¹(x).
Symmetry: The graph of an inverse function is a reflection of the original function's graph across the line y = x.
Composition rule: If f(x) and g(x) are inverses, then f(g(x)) = x and g(f(x)) = x.
One-to-one requirement: A function must be one-to-one (passing the horizontal line test) to have an inverse. If a function is not one-to-one, its domain must be restricted first.

How to Find the Inverse of a Function

Finding the inverse of a function follows a systematic process. Here is a reliable four-step method you can use every time:

Step 1: Replace f(x) with y

Write the function in the form y = ...

Step 2: Swap x and y

Interchange the variables so that every x becomes y and every y becomes x.

Step 3: Solve for y

Rearrange the equation to isolate y on one side.

Step 4: Replace y with f⁻¹(x)

The resulting expression is the inverse function.

Worked Examples

Example 1: Linear Function

Given: f(x) = 3x + 2

Step 1: y = 3x + 2

Step 2: Swap: x = 3y + 2

Step 3: Solve for y:

x - 2 = 3y
y = (x - 2) / 3

Step 4: f⁻¹(x) = (x - 2) / 3

So, if a test asks "which of the following is the inverse of f(x) = 3x + 2?", the correct answer would be f⁻¹(x) = (x - 2)/3.

Example 2: Simple Quadratic (with restricted domain)

Given: f(x) = x², where x ≥ 0

Step 1: y = x²

Step 2: Swap: x = y²

Step 3: Solve: y = √x

Step 4: f⁻¹(x) = √x

The domain restriction (x ≥ 0) is critical here. Without it, f(x) = x² would not be one-to-one, and no unique inverse would exist.

Example 3: Rational Function

Given: f(x) = 1 / (x - 1)

Step 1: y = 1 / (x - 1)

Step 2: Swap: x = 1 / (y - 1)

Step 3: Solve:

x(y - 1) = 1
xy - x = 1
xy = 1 + x
y = (1 + x) / x

Step 4: f⁻¹(x) = (1 + x) / x

Common Patterns in Multiple-Choice Questions

When you encounter a question that asks "which of the following is the inverse of...", the answer choices often include common mistakes. Here are the typical traps to watch out for:

Reciprocal confusion: A very common error is confusing f⁻¹(x) with 1/f(x). The superscript ⁻¹ does not mean a reciprocal — it means the inverse function Nothing fancy..
Incomplete algebraic solving: Some answer choices will show a partially solved equation where y has not been fully isolated.
Sign errors: Watch for plus/minus sign mistakes when rearranging terms Not complicated — just consistent..
Domain neglect: Some choices may represent the correct algebraic inverse but fail to include the necessary domain restriction.
Swapping error: Occasionally, a choice will present the original function itself, testing whether you actually performed the swap step.

How to Verify Your Answer

After finding the inverse, always verify using the composition method:

Compute f(f⁻¹(x)) — the result should equal x.
Compute f⁻¹(f(x)) — the result should also equal x.

Verification for Example 1:

f(f⁻¹(x)) = 3((x - 2)/3) + 2 = x - 2 + 2 = x ✓
f⁻¹(f(x)) = ((3x + 2) - 2)/3 = 3x/3 = x ✓

Both compositions return x, confirming the inverse is correct.

Special Cases and Tips

Functions That Are Their Own Inverses

Some functions are self-inverse, meaning f(f(x)) = x. The most common examples include:

f(x) = x (the identity function)
f(x) = -x
f(x) = 1/x, where x ≠ 0
f(x) = a - x, for any constant a

If you see any of these on a test, remember that their inverse is themselves Practical, not theoretical..

Exponential and Logarithmic Functions

Probably most frequently tested inverse pairs is:

f(x) = eˣ and f⁻¹(x) = ln(x)
f(x) = 10ˣ and f⁻¹(x) = log(x)

These are inverses of each