Inverse Function Calculator

An inverse function f⁻¹ undoes f: if f(a) = b, then f⁻¹(b) = a. For f(x) = 2x + 3, f(4) = 11 and f⁻¹(11) = 4, with f⁻¹(x) = (x − 3)/2.

• f⁻¹ swaps inputs and outputs: the domain of f is the range of f⁻¹, and the range of f is the domain of f⁻¹.
• The graph of f⁻¹ is the graph of f reflected in the line y = x: each point (a, b) becomes (b, a).
• Only one-to-one functions have inverses. f(x) = x² sends 3 and −3 to 9, so 9 has no single way back.
• f⁻¹(x) is not 1/f(x). For f(x) = 2x + 3, 1/f(x) = 1/(2x + 3), a different function.

To find the inverse of a function, write y = f(x), swap x and y, and solve the new equation for y. For f(x) = (3x − 2)/5:

1. Write y = (3x − 2)/5.
2. Swap x and y: x = (3y − 2)/5.
3. Multiply by 5: 5x = 3y − 2.
4. Add 2: 5x + 2 = 3y.
5. Divide by 3: y = (5x + 2)/3, so f⁻¹(x) = (5x + 2)/3.

Check with a number: f(4) = (12 − 2)/5 = 2, and f⁻¹(2) = (10 + 2)/3 = 4.

The steps undo the operations of f in reverse order: f multiplies by 3, subtracts 2 and divides by 5, so f⁻¹ multiplies by 5, adds 2 and divides by 3.

Type f(x) and the calculator returns f⁻¹(x) with every algebra step, a check and the mirror graph. The steps:

1. Choose Find f⁻¹(x) for a formula, f⁻¹ at a value for a number, or Check 2 functions to test a pair.
2. Type the function with x as the variable: 2x + 3, x^3, e^(2x), ln(x − 2), sqrt(x + 4) or (2x + 1)/(x − 3).
3. Read f⁻¹(x), the domain notes and a numeric check.
4. Open the mirror graph or the arrow diagram, and the worked steps.

Formulas come out when x appears once, or when f is linear, a ratio of 2 linear terms, or a quadratic. For anything else, the value mode solves f(x) = y numerically.

f and g are inverses when f(g(x)) = x and g(f(x)) = x for every x in their domains.

• f(x) = 2x + 3 and g(x) = (x − 3)/2: f(g(x)) = 2 · (x − 3)/2 + 3 = x, and g(f(x)) = (2x + 3 − 3)/2 = x. They are inverses.
• f(x) = eˣ and g(x) = ln x: e^(ln x) = x for x > 0, and ln(eˣ) = x for every x. They are inverses.
• f(x) = x² and g(x) = √x: g(f(−3)) = √9 = 3, not −3. They are inverses only after f is restricted to x ≥ 0.

Both compositions are needed. One of them can return x while the other fails, as with x² and √x.

A function has an inverse when every horizontal line crosses its graph at most once; that makes it one-to-one.

• x³ and eˣ pass: they always rise, so each output comes from one input.
• x² fails: the line y = 4 crosses at x = −2 and x = 2.
• sin x fails: the line y = 0.5 crosses infinitely often.
• |x| fails: |−3| = |3| = 3.

A function that only rises (or only falls) always passes. A positive derivative everywhere is a quick proof: f(x) = x³ + x has f′(x) = 3x² + 1 > 0, so it has an inverse even without a formula.

A function that fails the horizontal line test gets an inverse once its domain is cut to a one-to-one piece.

• x² on x ≥ 0 has inverse √x. On x ≤ 0 it has inverse −√x.
• x² + 4x + 3 = (x + 2)² − 1. On x ≥ −2 its inverse is √(x + 1) − 2.
• sin x on −π/2 ≤ x ≤ π/2 has inverse arcsin x; cos x on 0 ≤ x ≤ π has inverse arccos x.

The chosen piece becomes the range of the inverse. That is why arcsin returns angles from −90° to 90° only, and √x never returns a negative number.

For f(x) = (ax + b)/(cx + d), the inverse is f⁻¹(x) = (dx − b)/(a − cx); for a quadratic, complete the square and take one branch.

• (2x + 1)/(x − 3): swap to x = (2y + 1)/(y − 3), multiply out to xy − 3x = 2y + 1, collect y(x − 2) = 3x + 1, so f⁻¹(x) = (3x + 1)/(x − 2).
• The inverse has its vertical asymptote at x = a/c = 2, the horizontal asymptote of f.
• x² + 4x + 3: write (x + 2)² − 1, swap, solve: f⁻¹(x) = √(x + 1) − 2 on x ≥ −1.
• 5/(x + 1): x appears once, so undo the steps: f⁻¹(x) = 5/x − 1.

To find f⁻¹(b) without a formula, solve f(x) = b for x: f⁻¹(b) is the x that f sends to b.

• f(x) = x³ + x has no simple inverse formula, but f(2) = 10, so f⁻¹(10) = 2.
• Numerically: find 2 inputs where f(x) − b changes sign, then halve the gap until it is tiny (bisection).
• From a table: f⁻¹(b) is the x in the row where f(x) = b.
• The slope of the inverse is the reciprocal of the slope of f: (f⁻¹)′(10) = 1/f′(2) = 1/13 ≈ 0.0769.

Computer algebra systems invert functions by solving y = f(x) for x: Solve in Wolfram Language, solve in SymPy and finverse in MATLAB.

• Wolfram Language: InverseFunction[2 # + 3 &] or Solve[y == 2x + 3, x].
• SymPy: solve(Eq(y, 2*x + 3), x) returns [y/2 − 3/2].
• MATLAB: syms x; finverse(2*x + 3) returns x/2 − 3/2.
• GeoGebra: Inverse(2x + 3).
• TI-Nspire CAS: solve(y = 2x + 3, x).

These tools return every branch, so x² gives ±√y. Pick the branch that matches the domain you need.

The table pairs common functions with their inverses and the domain where each inverse works. Click a row to load the function into the calculator.

Exponentials and logarithms undo each other, powers and roots undo each other, and each trigonometric function is undone by its arc function on a restricted interval.

An inverse function answers the reverse question: given the result, what was the input?

• Temperature: F = 9C/5 + 32 turns Celsius into Fahrenheit; C = 5(F − 32)/9 turns it back. 98.6 °F is 37 °C.
• Taxi fare: cost = 3 + 2.5d. A $23 fare means d = (23 − 3)/2.5 = 8 km.
• Savings: A = 1000 · 1.05ᵗ. Doubling to $2000 takes t = ln 2 / ln 1.05 = 14.21 years.
• Sound: L = 10 log₁₀(I/I₀) in decibels; I = I₀ · 10^(L/10). 60 dB is 1,000,000 times I₀.
• Codes: decryption is the inverse of encryption, so an encryption map must be one-to-one.