Inverse Functions in Mathematics: Definition, Conditions, Computation, and Examples

1. What Is an Inverse Function?

An inverse function is a function that reverses the action of the original function. If a function f maps an input x to an output y, then the inverse function f⁻¹ maps y back to x.

f(x) = y
f⁻¹(y) = x

In simple terms, the inverse function undoes what the original function does.

2. Condition for the Existence of an Inverse

A function has an inverse if and only if it is one‑to‑one. This means no two different inputs may produce the same output.

Formal condition:

f(a) = f(b)  →  a = b

If a function is not one‑to‑one, its inverse will not be a function.

3. How to Compute an Inverse Function

To find the inverse of a function, follow these steps:

• Write the function as y = f(x)
• Swap x and y
• Solve the resulting equation for y
• Rewrite the result as f⁻¹(x)

Example:

f(x) = 3x - 1
y = 3x - 1
Swap x and y:
x = 3y - 1
Solve for y:
3y = x + 1
y = (x + 1) / 3
Thus:
f⁻¹(x) = (x + 1) / 3

4. Horizontal Line Test for Invertibility

A function is invertible if and only if every horizontal line intersects its graph at most once.

Example:

The function f(x) = x^3 passes the horizontal line test, so it is invertible.

5. Functions That Do Not Have Inverses

Many functions are not invertible because they are not one‑to‑one.

Example:

f(x) = x^2

Since f(2) = 4 and f(-2) = 4, the function is not invertible.

6. Making a Function Invertible by Restricting the Domain

Sometimes a function can become invertible by restricting its domain.

Example:

f(x) = x^2

If we restrict the domain to x ≥ 0, the function becomes one‑to‑one and therefore invertible:

f⁻¹(x) = √x

7. Relationship Between a Function and Its Inverse

A function and its inverse undo each other:

f(f⁻¹(x)) = x
f⁻¹(f(x)) = x

This is one of the most important properties of inverse functions.

8. Graph of an Inverse Function

The graph of an inverse function is symmetric with respect to the line y = x.

Example:

f(x) = 2x + 1
f⁻¹(x) = (x - 1) / 2

The graphs of these two functions are mirror images across the line y = x.

9. Conclusion

Inverse functions play a crucial role in mathematics because they allow us to recover the input from the output. Only one‑to‑one functions are invertible, and finding the inverse typically involves swapping x and y and solving for y. Understanding inverse functions is essential for calculus, algebra, and mathematical modeling.