This inverse function calculator finds f⁻¹(x) for a linear or simple rational function — the function that swaps the input and output roles.
Swap x and y, Then Solve — the Inverse Undoes the Function
Exponential and logarithmic functions are the classic inverse pair; the log side lives on our Logarithm Calculator. Part of our Math Calculators collection.
Worked Examples (Verified)
- f(x) = 2x + 6 (linear defaults): f⁻¹(x) = 0.5x − 3. Check: f(4) = 14, and f⁻¹(14) = 4.
- f(x) = (3x − 2)/(x + 4): f⁻¹(x) = (4x + 2)/(3 − x). The excluded value moves from x = −4 (original) to x = 3 (inverse).
- A horizontal line has no inverse function because many inputs share one output; restricting a domain can rescue an inverse, as with x² on x ≥ 0 → √x.
- Composition is the audit: an alleged inverse must return x in both orders, not just one convenient value.
Domain and Range Trade Places
- The inverse’s domain is the original’s range, and vice versa.
- Vertical asymptotes and horizontal asymptotes exchange roles under inversion.
- Notation warning: f⁻¹(x) is not 1/f(x); the superscript means inverse function, not reciprocal power.
- Only one-to-one functions have inverses on their full domains.
Related Tools
- Logarithm Calculator — inverse of exponentiation
- Exponent Calculator — powers and roots
- Math Calculators — the full collection
Frequently Asked Questions
How do you find the inverse of a function?
Write y = f(x), swap x and y, then solve for y. For f(x) = 2x + 6: x = 2y + 6 → y = (x − 6)/2 = 0.5x − 3. Verify with composition: applying f then f⁻¹ (or the reverse) returns the starting value.
What is the inverse of a rational function (ax+b)/(cx+d)?
f⁻¹(x) = (−dx + b)/(cx − a), provided ad − bc ≠ 0. Example: f(x) = (3x − 2)/(x + 4) has inverse (4x + 2)/(3 − x). If ad − bc = 0, the rational expression collapses toward a constant and is not invertible.
Does every function have an inverse?
No. Only one-to-one functions do on their full domain. Parabolas, absolute values and even powers fail the horizontal-line test unless their domains are restricted — x² on non-negative inputs inverts to √x.
Is f⁻¹(x) the same as 1/f(x)?
No. f⁻¹ denotes the inverse function that undoes f; 1/f(x) is the reciprocal. For f(x) = 2x, f⁻¹(x) = x/2 while 1/f(x) = 1/(2x) — different functions entirely.
How are the graphs related?
An inverse reflects the original graph across the line y = x because every point (a, b) becomes (b, a). Intercepts, asymptotes and domain/range endpoints swap coordinates accordingly.
Is this inverse function calculator free?
Yes — free, instant, no sign-up.