Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for precalculus inverse functions? Then this Precalculus lesson on Domains, Composition and Inverse Functions Explained explains each idea step by step, with worked examples.
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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