Function Composition Calculator
Compose linear and quadratic functions in both orders, simplify the resulting polynomials, and evaluate each composition at a chosen x value.
Function Composition Calculator inputs
Use the fields below for this exact calculation. Every visible assumption can be changed before the result is interpreted.
Outer function f(x)
Inner function g(x)
Function Composition Calculator results
How the function composition calculator works
The engine represents each function as polynomial coefficients and performs exact coefficient multiplication and addition during substitution.
Composition is substitution, not multiplication
A function composition calculator must keep the inner and outer functions separate. In f(g(x)), every occurrence of x in f is replaced by the complete expression g(x). Parentheses matter because all terms of the inner function are substituted before expansion. The calculator carries coefficient arrays through the substitution, expands products, combines like terms, and displays the simplified polynomial.
Why both orders are reported
Function composition is usually not commutative. That means f(g(x)) and g(f(x)) can have different coefficients, degrees, intercepts, and values even when the same two functions are used. This function composition calculator solves both directions side by side so a student can see the consequence of changing the order instead of assuming the answers are interchangeable.
Linear and quadratic combinations
A linear function composed with another linear function remains linear. A quadratic outer function with a linear inner function remains quadratic, while composing two quadratic functions can produce degree four. When a function type is set to linear, its quadratic coefficient is ignored. The degree cards make these changes explicit and help verify that the expanded answer has a plausible form.
Evaluating a composition correctly
There are two dependable evaluation paths. One path evaluates the inner function first and sends that number into the outer function. The other path simplifies the complete composition and then substitutes x. Both paths should agree. The calculator uses the simplified polynomial for its value card, while the displayed component functions let the visitor independently check the inner-then-outer route.
Domain and interpretation limits
Polynomial functions are defined for every real x, which keeps this version transparent and predictable. Rational, radical, logarithmic, trigonometric, piecewise, and inverse functions can introduce restricted domains or multiple branches and are not silently approximated here. A result from this function composition calculator applies to the entered linear or quadratic coefficients only.
Classroom and checking workflow
Start by writing f and g in standard form. Select the correct degree, enter every coefficient including zero, and choose an x value. Compare the displayed f(g(x)) and g(f(x)), then verify one result by evaluating the inner function first. Changing one coefficient at a time makes the live machine visual useful for exploring how composition changes structure and degree.
Function Composition Calculator: reliable-use checklist
Enter the exact model, not a nearby problem
The function composition calculator is deliberately limited to the rules described on this page. Confirm that the entered relationship, timing convention, domain, units, and event definition match the original problem. A numerically precise answer can still be inappropriate when the wrong model is selected.
Keep signs, order, and precision visible
Negative signs, function order, coordinate order, draw conditions, and beginning-versus-end timing can change the result materially. Preserve full precision during an independent check and round only the reported answer. The supporting cards are included so the main result is not copied without its assumptions.
Test a known example before relying on a custom case
Load the worked example and confirm that the displayed method behaves as expected. Then change one input at a time. This exposes sign errors, invalid ranges, and misunderstood settings more effectively than replacing every default simultaneously.
Save enough information to reproduce the answer
Keep the entered values, selected options, date, result cards, and any limitation shown in the detailed report. The function composition calculator runs locally and does not create an account or store a calculation history, so reproducibility depends on the record retained by the visitor.
Function Composition Calculator questions
Is f(g(x)) the same as f(x) times g(x)?
No. Composition substitutes the output of g into f; multiplication forms a product of the two function values.
Why can both evaluated answers match even when the formulas differ?
Two different polynomials can have the same value at one selected x. Compare the simplified formulas, not only one evaluation point.
What happens to coefficient a in linear mode?
It is ignored because a linear function has no x-squared term. The value remains available when switching back to quadratic mode.
Can this calculator prove two functions are inverses?
A useful inverse check is whether both compositions simplify to x. The displayed polynomials can reveal that for the supported linear and quadratic entries.