Enter two functions f(x) and g(x) and an optional x value to evaluate. The Function Composition Calculator computes (f∘g)(x) and (g∘f)(x), showing you both composed expressions and their numeric results at your chosen point. Also try the calculate Synthetic Division.
(f∘g)(x) Expression
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(g∘f)(x) Expression
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(f∘g)(x₀) Value
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(g∘f)(x₀) Value
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f(x₀) Value
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g(x₀) Value
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Function composition is an operation where you apply one function to the result of another. (f∘g)(x) means you first apply g to x, then apply f to that result — written as f(g(x)). The order matters: (f∘g)(x) and (g∘f)(x) generally produce different results. See also our calculate Unknown Variable Result, Constant of Proportionality (k) & Product Verification (x × y = k) — Inverse Variation.
(f∘g)(x) = f(g(x)) means g is applied first, then f. (g∘f)(x) = g(f(x)) means f is applied first, then g. These are usually not equal — function composition is not commutative in general.
Use standard mathematical notation with x as the variable. For example, enter x^2 + 1 for x² + 1, or 2*x + 3 for 2x + 3. You can also use functions like sqrt(x), abs(x), sin(x), cos(x), log(x), and exp(x).
Evaluating (f∘g)(x) at x = a gives you the numeric result of f(g(a)). For example, if g(x) = 2x + 3 and f(x) = x², then (f∘g)(2) = f(g(2)) = f(7) = 49. Enter your desired x value in the 'Evaluate at x' field. You might also find our System of Equations Calculator useful.
Yes — three or more functions can be composed, such as (f∘g∘h)(x) = f(g(h(x))). This calculator handles two functions at a time: f(x) and g(x). For three functions, you can compose two first, use the result as a new function, and compose again.
These two compositions are equal when f and g are inverses of each other — in that case, both compositions equal x. They can also be equal for specific function pairs, but it is not generally guaranteed.
The domain of (f∘g)(x) consists of all x values in the domain of g such that g(x) is also in the domain of f. In other words, you need g(x) to be defined, and then f must be defined at that resulting value.
Yes — function composition is associative. That means (f∘g)∘h = f∘(g∘h) for any three functions. However, it is not commutative, so f∘g ≠ g∘f in general.