Function Composition Calculator

Use * for multiplication, ^ for powers

Evaluate compositions at this value

What is function composition?

Function composition is the process of applying one function to the result of another. f(g(x)) means "evaluate g at x, then evaluate f at the result".

What does (f ∘ g)(x) mean?

The notation (f ∘ g)(x) is called "f circle g of x" and equals f(g(x)). The circle symbol ∘ represents composition. Read it right to left: apply g first, then f.

Is f(g(x)) the same as g(f(x))?

No! Function composition is NOT commutative. The order matters. f(g(x)) applies g first then f, while g(f(x)) applies f first then g. They usually give different results.

How do I find the domain of a composite function?

The domain of f(g(x)) includes all x in the domain of g such that g(x) is in the domain of f. Check restrictions from both functions.

Can I compose more than two functions?

Yes! You can compose any number of functions. For example, f(g(h(x))) applies h first, then g to that result, then f to that result. Work from the inside out.

What is function composition used for?

Function composition is fundamental in mathematics. It's used to build complex functions from simpler ones, in calculus (chain rule), and to model multi-step processes where one output becomes the next input.

How do I evaluate (f ∘ g)(5)?

First calculate g(5), then use that result as the input to f. If g(5) = 3, then (f ∘ g)(5) = f(g(5)) = f(3). Always work from the inside out.

What's the relationship between composition and inverse functions?

For inverse functions, f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. The composition of a function and its inverse gives the identity function, which just returns the input.