Inverse Function Calculator

Reverse a one-to-one function, track domain restrictions, and verify the result by composition.

Drag & drop or upload an image or PDF

Enter a function, such as f(x) = (3x - 5)/2

Name each function and the requested operation, such as f(x) = 2x + 3.

Try an example

An inverse function reverses inputs and outputs

If a function sends an input x to an output y, its inverse sends that output y back to x. This reversal is possible as a function only when every output comes from one input on the chosen domain.

The horizontal line test checks that one-to-one condition on a graph. Algebraically, finding the inverse swaps x and y and then solves for the new output. The domain of the inverse is the original range, and the inverse range is the original domain.

How to use the inverse function calculator

Enter the problem as written

Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.

Read the working, not only the answer

Each transformation is separated and explained so you can compare it with your own method.

Ask about any step

Continue in the same solution to request another method, check a restriction, or ask why a rule applies.

Composition check for inverse functions

Both identities must hold on the appropriate domains for the two functions to be inverses.

f^{- 1} (f (x)) = x, f (f^{- 1} (x)) = x

A reliable way to work through it

Check the one-to-one condition

Confirm that no output is produced by more than one input, or restrict the domain before finding an inverse.

Swap and solve

Write y = f(x), exchange x and y, and isolate y to obtain the inverse formula.

Verify by composition

Substitute each function into the other and simplify to x while respecting both domains.

Worked example

Find the inverse of f(x) = (3x - 5)/2

y = \frac{3 x - 5}{2}

Write the function output as y.

x = \frac{3 y - 5}{2}

Interchange x and y to reverse the input-output relationship.

2 x = 3 y - 5 ⟹ y = \frac{2 x + 5}{3}

Solve the swapped equation for y.

f (\frac{2 x + 5}{3}) = \frac{2 x + 5 - 5}{2} = x

Composition with the original function verifies the result.

The inverse is f^(-1)(x) = (2x + 5)/3, with all real numbers as its domain and range.

Common mistakes to check

Taking the reciprocal

The notation f^(-1) means inverse function, not 1/f(x).

Ignoring a repeated output

A function such as x squared needs a restricted domain before its inverse can be single-valued.

Forgetting domain and range

The inverse formula may be algebraically correct but incomplete without the branch and endpoint restrictions.

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Questions students ask

Does every function have an inverse function?

No. A function must be one-to-one on its domain, although a suitable domain restriction can sometimes make it invertible.

Is an inverse function the same as a reciprocal?

No. An inverse reverses input and output, while a reciprocal is the expression 1/f(x).

Why is the domain sometimes restricted?

Restricting the domain removes repeated outputs so the reversed relation assigns only one output to each input.

How can I verify an inverse?

Compose the original and inverse functions in both orders. Each composition should simplify to x on its valid domain.