ganak.appInverse Function Calculator
Free inverse function calculator: find the inverse of linear, rational, quadratic, exponential, logarithmic, square root, and cubic functions. Step-by-step solution, interactive graph, domain/range analysis, and verification.
f(x) = ax + b
f(x) = 3x + 5
f⁻¹(x) = (x − 5) / 3
Function & Inverse
Original and inverse expressions with domain swap
Domain & Range
Domain of f becomes range of f⁻¹ and vice versa
All non-constant linear functions are one-to-one and always invertible.
Step-by-Step Solution
How the inverse was found algebraically
Start with the function
y = 3x + 5
Swap x and y
x = 3y + 5
Isolate the y term
x − 5 = 3y
Solve for y
y = (x − 5) / 3
Write the inverse
f⁻¹(x) = (x − 5) / 3
Graph: f(x) vs f⁻¹(x)
The inverse is the reflection of f(x) over the line y = x
Verification: f(f⁻¹(x)) = x
Composing f with its inverse should return the original input
| x | f(x) | f⁻¹(f(x)) | Status |
|---|---|---|---|
| -2 | -1 | -2 | Verified |
| 0 | 5 | 0 | Verified |
| 1 | 8 | 1 | Verified |
| 3 | 14 | 3 | Verified |
| 5 | 20 | 5 | Verified |
What Is an Inverse Function?
Understanding the concept of function reversal
An inverse function “undoes” what the original function does. If f(x) maps an input x to an output y, then the inverse function f⁻¹(x) maps y back to x.
Key Relationship
f(f⁻¹(x)) = x and f⁻¹(f(x)) = x
Reflection
Mirror image over the line y = x
Domain Swap
Domain of f = Range of f⁻¹
Uniqueness
One-to-one functions have exactly one inverse
Inverse Formulas by Function Type
Quick reference for common function families
Linear
f(x) = ax + b
f⁻¹(x) = (x − b) / a
Rational
f(x) = (ax+b)/(cx+d)
f⁻¹(x) = (dx−b)/(−cx+a)
Quadratic
f(x) = ax² + bx + c
Restrict domain first, then use quadratic formula
Exponential
f(x) = a·bˣ + c
f⁻¹(x) = log_b((x−c)/a)
Logarithmic
f(x) = a·log_b(x−h)+k
f⁻¹(x) = b^((x−k)/a) + h
Square Root
f(x) = a√(x−h) + k
f⁻¹(x) = ((x−k)/a)² + h
Cubic
f(x) = ax³ + b
f⁻¹(x) = ³√((x−b)/a)
How to Find an Inverse Function
The standard algebraic procedure step by step
Replace f(x) with y
Write the function as y = f(x).
Swap x and y
Interchange the variables to get x = f(y).
Solve for y
Algebraically isolate y on one side.
Write the inverse
Replace y with f⁻¹(x).
Verify
Check that f(f⁻¹(x)) = x by composing.
Worked Examples
Linear: f(x) = 3x + 5
y = 3x + 5
x = 3y + 5
x − 5 = 3y
y = (x − 5) / 3
f⁻¹(x) = (x − 5) / 3
Rational: f(x) = (2x + 1)/(x − 1)
x = (2y + 1)/(y − 1)
x(y − 1) = 2y + 1
xy − 2y = x + 1
y(x − 2) = x + 1
f⁻¹(x) = (x + 1) / (x − 2)
Exponential: f(x) = 2ˣ
y = 2ˣ
x = 2ʸ
y = log₂(x)
f⁻¹(x) = log₂(x)
Square Root: f(x) = √(x − 3) + 1
x = √(y − 3) + 1
x − 1 = √(y − 3)
(x − 1)² = y − 3
f⁻¹(x) = (x − 1)² + 3
When Does an Inverse Exist?
The horizontal line test and one-to-one functions
A function has an inverse if and only if it is one-to-one (injective) — meaning no two different inputs produce the same output. The visual check is the horizontal line test.
Always Invertible
Linear (a ≠ 0), Exponential (b > 0, b ≠ 1), Logarithmic, Cubic (ax³ + b)
Invertible With Domain Restriction
Quadratic, Trigonometric — restrict domain to one monotone branch
Never Invertible
Constant functions f(x) = c — every input maps to the same output
Real-World Applications
Where inverse functions appear in practice
Cryptography
Encryption uses a function; decryption uses its inverse.
Unit Conversion
Celsius ↔ Fahrenheit conversions are inverse pairs.
Logarithmic Scales
pH, decibels, Richter scales use log/exp inverses.
Computer Science
Encoding/decoding, compression/decompression.
Economics
Supply/demand inverses find equilibrium prices.
Physics
Position ↔ time relationships via inverse functions.
Common Mistakes to Avoid
Frequent errors when finding inverse functions
Confusing f⁻¹(x) with 1/f(x)
The −1 superscript means inverse, not reciprocal. sin⁻¹(x) ≠ 1/sin(x).
Forgetting domain restriction
Quadratics need restricted domains to be one-to-one before inverting.
Sign errors when solving
Watch signs during cross-multiplication and distribution with rational functions.
Not verifying the answer
Always check f(f⁻¹(x)) = x and f⁻¹(f(x)) = x to confirm correctness.
Golden Rule
Always verify: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x
Frequently Asked Questions
Common questions about inverse functions and how to find them
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Last updated Apr 14, 2026