1. Check one-to-one behavior
A function needs exactly one input for each output on the chosen domain. A horizontal line must not meet its graph more than once.
f⁻¹(x) = 0.5x − 1.5
The original linear function is one-to-one.
Inverse Function Calculator with Steps
Get the inverse rule, original function, domain and range, one-to-one check, worked algebra, and an optional numerical verification.
Define the original function
Inverse result
Inverse function
Originalf(x) = 2x + 3
Inverse domainAll real numbers
Inverse rangeAll real numbers
Evaluationf⁻¹(11) = 4
Composition checkf(4) = 11 ✓
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Step-by-step solution
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How to find an inverse function
2. Exchange input and output
Write y=f(x), then interchange x and y. This reverses every ordered pair.
3. Isolate y
Undo the original operations in reverse order. The isolated expression is f⁻¹(x).
4. Swap domain and range
The original range becomes the inverse domain, while the original domain becomes the inverse range.
Composition identity
f(f⁻¹(x)) = x and f⁻¹(f(x)) = x
These identities apply only where each composition is defined. The superscript −1 names the inverse under composition; it does not mean the reciprocal 1/f(x).
Supported inverse forms
Linear
ax+b reverses to (x−b)/a, provided a≠0.
Shifted power
a(x−h)ⁿ+k reverses with an nth root. Even powers require a left or right branch.
Rational
(ax+b)/(cx+d) reverses to (b−dx)/(cx−a) when ad−bc≠0.
Exponential and logarithmic
These families reverse each other. The base must be positive and cannot equal 1.
Accuracy, privacy, and scope
The symbolic inverse is formed directly from the algebraic parameters. Optional evaluations use JavaScript floating-point arithmetic and display up to 12 significant digits. A successful numerical composition check is evidence for that test value, while the algebraic steps establish the general rule.
Inputs must be finite numbers with absolute value no greater than 10¹². Extremely large exponential results are rejected instead of displaying infinity. This calculator covers the five templates shown; it does not parse an arbitrary expression, automatically choose a restricted domain, or claim that a non-one-to-one relation has an inverse function. Last reviewed: 4 August 2026.
Inverse function FAQs
What is an inverse function?
An inverse reverses the original input-output rule. If f(a)=b, then f⁻¹(b)=a, as long as the values lie in the stated domains.
How do I find an inverse?
Set y=f(x), interchange x and y, solve for y, and state the new domain. Always check that the original function is one-to-one on its domain.
Why does an even power need a domain restriction?
An even-power graph repeats output values on opposite sides of its turning point. Selecting x≥h or x≤h keeps one monotonic branch and makes the inverse single-valued.
Is f⁻¹(x) the same as 1/f(x)?
No. f⁻¹ reverses a function under composition. The expression 1/f(x) is its reciprocal and usually has a different graph, domain, and rule.
What happens to domain and range?
They swap. Every permitted output of the original becomes a permitted input of the inverse, and every original input becomes an inverse output.
How can I verify the result?
Compose the functions. Both f(f⁻¹(x)) and f⁻¹(f(x)) should simplify to x on the appropriate domains. You can also use the optional evaluation field for a numerical check.