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
xandy - 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.