Inverse Function Calculator

About Inverse Function Calculator

Welcome to our Inverse Function Calculator, a free online tool that helps you find the inverse of a function with detailed step-by-step instructions. Whether you are a student learning about inverse functions, preparing for calculus, or a teacher creating examples, this calculator provides clear explanations of the algebraic process.

What is an Inverse Function?

An inverse function, denoted as $f^{-1}(x)$, reverses the operation of the original function $f(x)$. If $f(a) = b$, then $f^{-1}(b) = a$. In other words, the inverse function "undoes" what the original function does.

The key properties of inverse functions include:

• Composition property: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$
• Graphical relationship: The graph of $f^{-1}(x)$ is the reflection of $f(x)$ across the line $y = x$
• Domain-range swap: The domain of $f$ becomes the range of $f^{-1}$, and vice versa

How to Find the Inverse of a Function

Follow these steps to find the inverse function algebraically:

Step 1: Replace f(x) with y

Start by writing the function as $y = f(x)$. This makes the algebraic manipulation easier.

Step 2: Swap x and y

Interchange the variables x and y in the equation. This reverses the input-output relationship.

Step 3: Solve for y

Use algebraic techniques to isolate y on one side of the equation. This is often the most challenging step.

Step 4: Write in Function Notation

Replace y with $f^{-1}(x)$ to express the inverse function properly.

Step 5: Verify (Optional)

Confirm your answer by checking that $f(f^{-1}(x)) = x$.

Common Inverse Functions

Original Function $f(x)$	Inverse Function $f^{-1}(x)$
$f(x) = x + a$	$f^{-1}(x) = x - a$
$f(x) = ax$	$f^{-1}(x) = \frac{x}{a}$
$f(x) = ax + b$	$f^{-1}(x) = \frac{x - b}{a}$
$f(x) = x^2$ (for $x \geq 0$)	$f^{-1}(x) = \sqrt{x}$
$f(x) = x^3$	$f^{-1}(x) = \sqrt[3]{x}$
$f(x) = e^x$	$f^{-1}(x) = \ln(x)$
$f(x) = \ln(x)$	$f^{-1}(x) = e^x$
$f(x) = \frac{1}{x}$	$f^{-1}(x) = \frac{1}{x}$

When Does a Function Have an Inverse?

Not all functions have inverse functions. A function has an inverse if and only if it is one-to-one (also called injective). This means each output value corresponds to exactly one input value.

The Horizontal Line Test

A function passes the horizontal line test if no horizontal line intersects its graph more than once. If a function passes this test, it has an inverse.

• Linear functions (with non-zero slope) are always one-to-one
• Quadratic functions are not one-to-one over all real numbers (they fail the horizontal line test)
• Strictly monotonic functions (always increasing or always decreasing) are one-to-one

Restricting the Domain

When a function is not one-to-one, we can restrict its domain to make it one-to-one. For example:

• $f(x) = x^2$ is not one-to-one, but $f(x) = x^2$ for $x \geq 0$ is one-to-one with inverse $f^{-1}(x) = \sqrt{x}$
• $f(x) = \sin(x)$ is not one-to-one, but $f(x) = \sin(x)$ for $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$ is one-to-one with inverse $f^{-1}(x) = \arcsin(x)$

Examples

Example 1: Linear Function

Find the inverse of $f(x) = 3x - 5$

Solution:

1. Write as $y = 3x - 5$
2. Swap: $x = 3y - 5$
3. Solve for y: $x + 5 = 3y$, so $y = \frac{x + 5}{3}$
4. Therefore, $f^{-1}(x) = \frac{x + 5}{3}$

Example 2: Rational Function

Find the inverse of $f(x) = \frac{x - 1}{x + 2}$

Solution:

1. Write as $y = \frac{x - 1}{x + 2}$
2. Swap: $x = \frac{y - 1}{y + 2}$
3. Solve: $x(y + 2) = y - 1$, so $xy + 2x = y - 1$
4. Rearrange: $xy - y = -1 - 2x$, so $y(x - 1) = -2x - 1$
5. Therefore, $f^{-1}(x) = \frac{-2x - 1}{x - 1} = \frac{2x + 1}{1 - x}$

Tips for Using This Calculator

• Enter functions using x as the variable
• Use * for multiplication (e.g., 2*x instead of 2x)
• Use ^ or ** for exponents (e.g., x^2 or x**2)
• Use sqrt(x) for square root
• Use log(x) for natural logarithm
• Use exp(x) or e^x for exponential function

Frequently Asked Questions

What does the -1 in f^(-1)(x) mean?

The -1 in $f^{-1}(x)$ is not an exponent. It is notation that indicates the inverse function. It should not be confused with $\frac{1}{f(x)}$, which is the reciprocal of f(x).

Can I find the inverse of any function?

Not all functions have inverses. Only one-to-one functions have inverse functions. If a function fails the horizontal line test, it does not have an inverse over its entire domain, but you may be able to restrict the domain to create an invertible function.

How do I verify that my inverse is correct?

To verify, check that both $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. If both compositions equal x, your inverse is correct.

Additional Resources

To learn more about inverse functions:

• Inverse Function - Wikipedia
• Inverse Functions - Khan Academy
• Inverse Function - Wolfram MathWorld