Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Quick answer: The domain of a function is the set of allowed inputs, composition feeds one function’s output into another, and an inverse function undoes the original by swapping inputs and outputs. A function has an inverse only when it is one-to-one, which restricting the domain can guarantee.
Standards: F-IF.A.1, F-BF.A.1c, F-BF.B.4b, F-BF.B.4c, F-BF.B.4d · Level: Grade 12 Precalculus
How do you find the domain of a function?
The domain is the set of inputs a function can accept. For real-valued functions, only two things cause trouble: dividing by zero and taking an even root of a negative number. Write the answer in interval notation, where a square bracket keeps an endpoint and a parenthesis leaves it out.
- Domain: every allowed input; range: every output that actually occurs.
- One-to-one: no two inputs share an output (passes the horizontal line test).
- Composition: (f ∘ g)(x) = f(g(x)).
- Inverse: the function that undoes f(x), written f−1(x).
For f(x) = √(x − 3) ÷ (x − 7), the root needs x − 3 ≥ 0, so x ≥ 3, and the denominator forbids x = 7. The domain is [3, 7) ∪ (7, ∞).
How do you compose and decompose functions?
To evaluate f(g(x)), work from the inside out. The order matters: with g(x) = 2x + 3 and f(x) = x2, we get (f ∘ g)(x) = (2x + 3)2, but (g ∘ f)(x) = 2x2 + 3. To decompose h(x) = √(5x − 1), name the inner piece 5x − 1 and the outer piece √u. A composite is defined only where x is in the domain of the inner function and its output is in the domain of the outer one.
How do you find an inverse function?
- Write y = f(x).
- Swap x and y.
- Solve the new equation for y.
- Write f−1(x) = that expression and check that f(f−1(x)) = x.
Let f(x) = 2x + 1x − 3. Swapping gives x = 2y + 1y − 3, so xy − 3x = 2y + 1 and y(x − 2) = 3x + 1. Therefore f−1(x) = 3x + 1x − 2. Domain and range trade places.
Tables and graphs work the same way: if f(3) = 8, then f−1(8) = 3. A function that fails the horizontal line test has no inverse until its domain is restricted, for example x2 on x ≥ 0.
f−1(x) does not mean 1/f(x). The little −1 labels the inverse function; it is not an exponent.
Test your inverse by plugging a number through both functions. If you do not land back where you started, an algebra step slipped.
- Exclude zero denominators and negative numbers under even roots when finding a domain.
- (f ∘ g)(x) = f(g(x)): evaluate the inner function first; order matters.
- To invert a function, swap x and y, then solve for y.
- The graph of an inverse is the reflection of the original across the line y = x.
- If f(a) = b then f⁻¹(b) = a, and f(f⁻¹(x)) = x.
Domains, Composition and Inverse Functions Explained: frequently asked questions
What is the domain of a function in interval notation?
It lists every allowed input as one or more intervals. Use brackets for included endpoints, parentheses for excluded ones, and the union symbol ∪ to join separate pieces, such as (−∞, 2) ∪ (2, ∞).
Is f(g(x)) the same as g(f(x))?
Usually not. The inner function acts first, so swapping the order changes the result. For f(x) = x + 5 and g(x) = 3x, f(g(x)) = 3x + 5 but g(f(x)) = 3x + 15.
How can you tell whether a function has an inverse?
Apply the horizontal line test: if any horizontal line crosses the graph twice, two inputs share an output and the function is not one-to-one. Restrict the domain to a one-to-one piece to create an inverse.
Is the inverse f⁻¹(x) equal to 1/f(x)?
No. The reciprocal 1/f(x) divides 1 by the output, while f⁻¹(x) undoes the function. For f(x) = 2x, the inverse is x/2 and the reciprocal is 1/(2x).
How do you restrict a domain to make a function invertible?
Keep one side of the turning point. For f(x) = (x − 4)² + 1, choosing x ≥ 4 makes the graph strictly increasing, so every output has exactly one input.
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