Function Composition

Chain functions together — feed one output into another input

Function composition means feeding the output of one function into another. If f(x) = x² and g(x) = x + 3, then f(g(x)) = (x + 3)² — first add 3, then square.

Order matters: f(g(x)) ≠ g(f(x)) in general. With the same functions, g(f(x)) = x² + 3 — first square, then add 3. The graph makes the difference obvious.

Ask the AI anything — try "What is f(g(2))?" or "Decompose h(x) = sqrt(2x + 1)."

Related Topics

What is a Function?Inverse FunctionsFunction Transformations

Suggested Lessons

Slope Quadratic Equations Unit Circle Word Problems
What is function composition?
Function composition is the process of applying one function to the result of another. Written as (f ∘ g)(x) = f(g(x)), it means: first apply g to x, then apply f to the result.
Does the order of composition matter?
Yes, almost always. f(g(x)) and g(f(x)) are usually different functions. For example, squaring then adding 3 gives a different result than adding 3 then squaring.
How do I decompose a composite function?
Look for an "inner" and "outer" operation. For h(x) = √(2x + 1), the inner function is g(x) = 2x + 1 and the outer function is f(x) = √x, so h = f(g(x)).
What are real-world examples of composition?
Converting Celsius to Fahrenheit then to a gas bill amount, applying a discount then sales tax, or zooming in on a map then rotating — any time you chain two processes together.
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.