Composite Function Calculator
Find and evaluate the composite functions f(g(x)) and g(f(x)) using two polynomial-style functions. This calculator supports linear and quadratic function forms, shows the substitution process, and evaluates both compositions at a chosen x-value.
Enter your functions
This version uses quadratic-style input for both functions: f(x) = ax² + bx + c and g(x) = dx² + ex + f. Linear functions also work by setting the x² coefficient to 0.
f(x) coefficients
g(x) coefficients
What this calculator returns
f(g(x)), g(f(x)), both evaluated values at your chosen x, and substitution steps for the compositions.
Best use cases
Use it for algebra class, function composition review, homework checks, and polynomial substitution practice.
How to use this composite function calculator
- Enter the coefficients for f(x) and g(x).
- Enter the x-value where you want both compositions evaluated.
- Click Calculate to build f(g(x)) and g(f(x)).
- Review the evaluated results and substitution steps.
Composite function idea
Composing functions means taking the output of one function and using it as the input of another function.
g(f(x)) means plug f(x) into g
Why order matters
In most cases, f(g(x)) and g(f(x)) are not the same. Function composition depends on order, so switching which function goes first often changes the result.
FAQ
What is a composite function?
A composite function is a function formed by putting one function inside another.
Is f(g(x)) always the same as g(f(x))?
No. In general, the order matters, so the two compositions are usually different.
Can I use linear functions here?
Yes. Just set the x² coefficient to 0 and the function becomes linear.
Does this simplify to a standard polynomial form?
This version shows clean substituted expressions and evaluated values, which is often enough for homework and checking logic quickly.