Enter f(x) and get its inverse f⁻¹(x) with every algebra step. When f is not one-to-one, the calculator restricts the domain for you, gives the matching domain and range of the inverse, and plots f and f⁻¹ as mirror images in the line y = x.
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Examples:
Use x as the variable: + − * / ^, sqrt, cbrt, abs or |x|, ln, log, e^x, sin, cos, tan, asin, acos, atan. Implicit multiplication like 2x works.
Inverse function
Domain of f
Range of f = domain of f⁻¹
Range of f⁻¹ = domain of f
Steps: swap x and y, then solve for y
Graph: f, f⁻¹ and y = x
f(x)f⁻¹(x)y = x
Check: f⁻¹(f(x)) = x
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How to use the Inverse Function Calculator
Type the function f(x), for example (2x + 1)/(x - 3), x^2 + 4x + 1 or e^(2x) + 1, or tap an example.
Press Find inverse (or Enter).
Read f⁻¹(x) at the top, with the domain and range of f and of its inverse.
Pick a branch if asked: functions like x² are not one-to-one, so choose which half of the domain to keep.
Follow the steps and check the graph: f and f⁻¹ are reflections of each other in the dashed line y = x.
Inverses of common functions
f(x)
f⁻¹(x)
Restriction needed on f
Domain of f⁻¹
mx + b (m ≠ 0)
(x − b)/m
None
All reals
(ax + b)/(cx + d)
(dx − b)/(a − cx)
None (if ad − bc ≠ 0)
x ≠ a/c
x²
√x
x ≥ 0
x ≥ 0
a(x − h)² + k, a > 0
h + √((x − k)/a)
x ≥ h
x ≥ k
x³
∛x
None
All reals
√x
x²
None
x ≥ 0
eˣ
ln x
None
x > 0
bˣ
log_b x = ln x / ln b
None
x > 0
ln x
eˣ
None
All reals
sin x
arcsin x
−π/2 ≤ x ≤ π/2
[−1, 1]
cos x
arccos x
0 ≤ x ≤ π
[−1, 1]
tan x
arctan x
−π/2 < x < π/2
All reals
|x|
x
x ≥ 0
x ≥ 0
Undo the operations in reverse order
Solving x = f(y) for y is the same as undoing what f does, last step first. Read f(x) = 2·e3x − 5 as: multiply by 3, take e to that power, double it, subtract 5. The inverse runs backwards: add 5, halve, take ln, divide by 3, so f⁻¹(x) = ln((x + 5)/2)/3. Each "undo" is one line in the step list: subtracting undoes adding, dividing undoes multiplying, a root undoes a power, a logarithm undoes an exponential, and arcsin undoes sin.
Checking your answer
Composition: f(f⁻¹(x)) = x for every x in the domain of f⁻¹, and f⁻¹(f(x)) = x for every x in the domain of f. The check table runs the second test with real numbers.
Points: if f(2) = 5, then f⁻¹(5) must be 2.
Graph: the two curves must be mirror images in y = x. If they cross, they cross on that line (or in symmetric pairs).
Domains: the domain of your formula for f⁻¹ should match the range of f. If it is larger, the formula only applies on part of it.
Frequently asked questions
How do you find the inverse of a function?
Write y = f(x), swap x and y to get x = f(y), and solve that equation for y. The result is f⁻¹(x). For f(x) = 3x − 7: x = 3y − 7, so 3y = x + 7 and f⁻¹(x) = (x + 7)/3.
Does every function have an inverse?
Only one-to-one functions have an inverse function: each output must come from exactly one input. A quick test is the horizontal line test: if any horizontal line crosses the graph twice, the function is not one-to-one. f(x) = x² fails, but restricted to x ≥ 0 it passes and its inverse is √x.
How do you find the inverse of a quadratic function?
Complete the square first. For f(x) = x² + 4x + 1 = (x + 2)² − 3, swapping gives x = (y + 2)² − 3, so y + 2 = ±√(x + 3). Choosing the branch x ≥ −2 keeps the + sign: f⁻¹(x) = √(x + 3) − 2, with domain x ≥ −3.
How do you find the inverse of a rational function?
For f(x) = (ax + b)/(cx + d), swap to x = (ay + b)/(cy + d), multiply by (cy + d), collect the y terms and factor: y(cx − a) = b − dx. So f⁻¹(x) = (dx − b)/(a − cx). Example: (2x + 1)/(x − 3) has inverse (3x + 1)/(x − 2).
How are the domain and range of f and f⁻¹ related?
They swap: the domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f. For f(x) = eˣ (domain all reals, range (0, ∞)), the inverse ln x has domain (0, ∞) and range all reals.
Why is the graph of the inverse a reflection in y = x?
If (a, b) is on the graph of f, then f(a) = b and f⁻¹(b) = a, so (b, a) is on the graph of f⁻¹. Swapping coordinates is the same as reflecting across the line y = x.
Is f⁻¹(x) the same as 1/f(x)?
No. f⁻¹ undoes f (f⁻¹(f(x)) = x), while 1/f(x) is the reciprocal of the output. For f(x) = 2x the inverse is x/2, but the reciprocal is 1/(2x).
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