How do you find the inverse of a function?

Write the function as y = f(x), swap x and y, then solve the new equation for y. The final y-expression is f⁻¹(x). After finding it, check whether the original function is one-to-one and verify by composing f(f⁻¹(x)) or f⁻¹(f(x)).

What does one-to-one mean?

A function is one-to-one if no two different input values produce the same output value. This condition is important because an inverse function must give one output for each input. If the original function repeats y-values, its inverse relation will not pass the vertical line test.

Why does a quadratic need a restricted domain?

A full quadratic graph is a parabola, and most horizontal lines hit it twice. That means the function is not one-to-one. By restricting the domain to one side of the vertex, such as x ≥ h or x ≤ h, you keep only one branch of the parabola and make an inverse function possible.

How do I check an inverse function?

Compose the original function with the inverse. If f(f⁻¹(x)) simplifies to x and f⁻¹(f(x)) also simplifies to x over the allowed domains, the inverse is correct. For restricted functions, make sure the domain and range restrictions are also respected.

Is the inverse of f(x) the same as 1/f(x)?

No. This is a common notation mistake. f⁻¹(x) means the inverse function, not the reciprocal. The reciprocal of f(x) is 1/f(x). These are completely different ideas unless a very special function makes them coincide.

Education

Inverse Function Calculator

Find inverse functions by swapping x and y, solving back for y, and checking whether the original function is one-to-one.

Find inverse functions for linear, fractional linear, or restricted quadratic functions. The calculator also explains when an inverse exists.

Function type

b / h

c / k

Quadratic branch

Inverse function

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Original function—

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What an inverse function really does

An inverse function reverses the original function. If f sends 4 to 9, then f⁻¹ sends 9 back to 4. OpenStax explains inverse functions as functions that undo another function in its inverse functions section, including the need for a function to be one-to-one.

The one-to-one condition matters because an inverse function cannot send one input to two different outputs. That is why ordinary linear functions with nonzero slope have inverses, but a full quadratic needs a restricted domain before it can have an inverse function.

Worked example: inverse of a linear function

f(x) = 2x + 3\n\nStep 1: y = 2x + 3\nStep 2: swap x and y: x = 2y + 3\nStep 3: solve for y: x - 3 = 2y\n\nf⁻¹(x) = (x - 3)/2

You can check by composing: f(f⁻¹(x)) should simplify back to x.

Why some functions do not have an inverse function

A function can fail to have an inverse if two different x-values produce the same y-value. For example, f(x)=x² sends both 2 and -2 to 4. Unless you restrict the domain to one side, the inverse would have to send 4 to both 2 and -2, which is not a function.

Common questions

Write the function as y = f(x), swap x and y, then solve the new equation for y. The final y-expression is f⁻¹(x). After finding it, check whether the original function is one-to-one and verify by composing f(f⁻¹(x)) or f⁻¹(f(x)).
A function is one-to-one if no two different input values produce the same output value. This condition is important because an inverse function must give one output for each input. If the original function repeats y-values, its inverse relation will not pass the vertical line test.
A full quadratic graph is a parabola, and most horizontal lines hit it twice. That means the function is not one-to-one. By restricting the domain to one side of the vertex, such as x ≥ h or x ≤ h, you keep only one branch of the parabola and make an inverse function possible.
Compose the original function with the inverse. If f(f⁻¹(x)) simplifies to x and f⁻¹(f(x)) also simplifies to x over the allowed domains, the inverse is correct. For restricted functions, make sure the domain and range restrictions are also respected.
No. This is a common notation mistake. f⁻¹(x) means the inverse function, not the reciprocal. The reciprocal of f(x) is 1/f(x). These are completely different ideas unless a very special function makes them coincide.