An inverse function undoes another function: if f sends a to b, its inverse f−1 sends b back to a. For f(x) = 2x + 3, the inverse is f−1(x) = x − 32. Only one-to-one functions have inverses, and the graph of f−1 is the mirror image of f across y = x.
Every point (a, b) on f has a partner (b, a) on f−1, so the two graphs are reflections of each other across y = x.
Formal definition and notation
Let f be a one-to-one function with domainA and range B. Its inverse is the function f−1 with domain B and range A such that f−1(y) = x exactly when f(x) = y. Equivalently, f−1(f(x)) = x for every x in A and f(f−1(y)) = y for every y in B. The −1 is notation, not an exponent: f−1(x) ≠ 1f(x) in general.
The inverse reverses the steps in reverse order: f multiplies by 2 then adds 3, so f−1 subtracts 3 then divides by 2.
Inputs and outputs trade places: if (4, 11) is on f, then (11, 4) is on f−1.
The domain of f−1 is the range of f, and the range of f−1 is the domain of f.
A function has an inverse only if it is one-to-one: it passes the horizontal line test.
f(x) = x2 is not one-to-one, but restricted to x ≥ 0 it has the inverse √x.
Exponential and logarithmic functions are inverses: y = 10x and y = logx.
How to find an inverse
How to find an inverse
y = f(x) ⇒ swap x, y ⇒ solve for y
x
after the swap, the input of the inverse (an output of f)
y
after solving, the formula for f−1(x)
Check the answer by composing: both f(f−1(x)) and f−1(f(x)) must simplify to x.
Try it: run the inverse
Interactive
This machine is f−1 for f(x) = 3x + 2: it subtracts 2, then divides by 3. Feed it the outputs of f for 1, 2 and 4, which are 5, 8 and 14, and you get 1, 2 and 4 back. Hide the rule and see if a friend can find it.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
The −1 in f−1 names the inverse; it is not an exponent. 1f(x) is the reciprocal, written (f(x))−1.
✗ MistakeInverse of f(x) = 2x + 3: x2 − 3.
✓ Insteadf−1(x) = x − 32.
Undo the steps in reverse order: f doubles then adds 3, so the inverse subtracts 3 first, then halves. Check: f(4) = 11, and 11 − 32 = 4, but 112 − 3 = 2.5.
✗ MistakeThe inverse of f(x) = x2 is √x for all x.
✓ InsteadOnly after restricting f to x ≥ 0.
f(3) = f(−3) = 9, so “undo 9” has two answers. A function must be one-to-one to have an inverse.
Not to be confused with
Inverse Functionvs.Reciprocal
The reciprocal of the output is 1f(x); the inverse function reverses the input-output pairs. For f(x) = x + 1: reciprocal 1x + 1, inverse x − 1.
Inverse Functionvs.Inverse operations
Inverse operations are single steps that undo each other, like adding 3 and subtracting 3. An inverse function undoes a whole function, using inverse operations in reverse order.
x−1 means 1x, a reciprocal. f−1 means inverse function. Same symbol, different meanings.
Quick check: 3 questions
Question 1 of 3
What is the inverse of f(x) = 5x + 1?
Show answer and explanation
Answer: D. Swap and solve: x = 5y + 1, so y = x − 15. The first choice undoes the steps in the wrong order, the second is the reciprocal, and the third only undoes the 1.
Question 2 of 3
If f(7) = 2, what is f−1(2)?
Show answer and explanation
Answer: A. The inverse sends each output back to its input: f takes 7 to 2, so f−1 takes 2 to 7. The fractions treat −1 as an exponent.
Question 3 of 3
Which function does NOT have an inverse over all real numbers?
Show answer and explanation
Answer: B.x2 + 1 gives the same output for x and −x (both 2 and −2 give 5), so it is not one-to-one. The other three always increase or always decrease, so each output comes from one input.
Inverse Function: frequently asked questions
How do you find the inverse of a function?
Replace f(x) with y, swap x and y, and solve the new equation for y. That y is f−1(x). For f(x) = x3 + 4: swap to x = y3 + 4, subtract 4 and multiply by 3 to get f−1(x) = 3x − 12. Then check that f(f−1(x)) = x.
Does every function have an inverse?
No. Only one-to-one functions do, where each output comes from exactly one input. On a graph, every horizontal line must cross the graph at most once: the horizontal line test. A parabola fails it, so y = x2 has no inverse unless you keep only one half, such as x ≥ 0.
How are the graphs of a function and its inverse related?
They are reflections of each other across the line y = x. Swapping the coordinates of every point, as the inverse does, flips the picture over that diagonal. If the graph of f passes through (1, 5), the graph of f−1 passes through (5, 1). Any point where they meet on that line is unchanged.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSF-BF.B.4, CCSS HSF-BF.B.4.a
Cite this entry
Effortless Math Team. “Inverse Function.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/inverse-function/
Flashcard
TermInverse Functionf−1(f(x)) = x
DefinitionThe function that undoes f, swapping inputs and outputs. For f(x) = 2x + 3, f−1(x) = x − 32.
DomainThe domain of a function is the set of all input values (usually x-values) the function can accept. Unless a problem says otherwise, it is every real number except values that would divide by zero or take an even root of a negative number. On a graph, the domain is how far the graph reaches left and right.Functions · Grade 8, Algebra 1, Algebra 2ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8ReciprocalThe reciprocal of a number is 1 divided by that number, so the two multiply to 1. For a fraction, swap the numerator and denominator: the reciprocal of 34 is 43, and the reciprocal of 5 is 15. Zero has no reciprocal. Dividing by a fraction is the same as multiplying by its reciprocal.Number System · Grade 5, Grade 6, Grade 7Cube RootThe cube root of a number is the value that, used as a factor three times, gives that number. It is written 3√x. Because 4 × 4 × 4 = 64, 3√64 = 4. Unlike square roots, cube roots of negative numbers are real numbers: 3√−27 = −3.Number System · Grade 8, Algebra 1, Algebra 2Inverse OperationsInverse operations are operations that undo each other. Addition and subtraction are inverses: 9 + 6 − 6 = 9. Multiplication and division are inverses: 7 × 4 ÷ 4 = 7. You use them to check answers and to solve equations, undoing each step in reverse order until the variable is alone.Operations · Grade 1, Grade 3, Grade 6Negative ExponentA negative exponent tells you to take the reciprocal of the base raised to the matching positive exponent: a−n = 1an for a ≠ 0. So 2−3 = 18 and 10−2 = 0.01. A negative exponent makes a number smaller in size, but it never makes the value negative.Algebra · Grade 8, Algebra 1ParabolaA parabola is the U-shaped curve that is the graph of a quadratic function such as y = x2. It is symmetric about a line called the axis of symmetry, and its turning point is the vertex. Geometrically, it is every point equally far from a fixed point (the focus) and a fixed line (the directrix).Functions · Algebra 1, Algebra 2, GeometryReflectionA reflection is a transformation that flips a figure over a line to make a mirror image. That line is the line of reflection: each point and its image are the same distance from the line, on opposite sides. The image is congruent but faces the other way. Across the y-axis, (x, y) goes to (−x, y).Geometry · Grade 8, GeometryFunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1, Algebra 2Domain and RangeDomain and range describe a function: the domain is the set of all inputs it accepts, and the range is the set of all outputs it produces. For the points (1, 4) and (2, 5), the domain is {1, 2} and the range is {4, 5}. Restrictions usually come from division by zero or square roots of negatives.Functions · Grade 8, Algebra 1, Algebra 2Composition of FunctionsA composition of functions applies one function to the output of another. The composite f(g(x)), also written (f ∘ g)(x), means: find g(x) first, then put that result into f. If g(x) = x + 1 and f(x) = x2, then f(g(2)) = f(3) = 9. Order matters.Functions · Algebra 2, PrecalculusLogarithmA logarithm is the exponent you need to raise a base to in order to get a given number. logbx = y means by = x, so log2 32 = 5 because 25 = 32. A log with no base written uses base 10: log 1000 = 3. Logs undo exponents.Functions · Algebra 2, PrecalculusExponential FunctionAn exponential function is a function in which the input is the exponent: f(x) = a · bx, with a ≠ 0, base b > 0 and b ≠ 1. Each time x goes up by 1, the output is multiplied by the same factor b. If b > 1 the function grows; if 0 < b < 1 it decays.Functions · Algebra 1, Algebra 2, PrecalculusVertical Line TestThe vertical line test is a way to tell from a graph whether it shows a function: if any vertical line crosses the graph more than once, it is not a function. If every vertical line crosses it at most once, each input x has only one output y, so the graph is a function.Functions · Grade 8, Algebra 1, Algebra 2Square RootA square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.Number System · Grade 8, Algebra 1