Composite Function Calculator

Composition of Functions Calculator two functions instantly. Enter f(x) and g(x) to see f∘g and g∘f with simplified expressions and clear steps displayed.

Function f(x) Use x as the variable. Built-ins: sin, cos, exp, log, sqrt, etc. Multiplication may need * (e.g., 2*x).

Function g(x) Type any valid expression in x. Powers use ^ (e.g., x^2). No extra fields needed—just f(x) and g(x).

Equation Preview

Given f(x)=2*x+3, g(x)=x^2 → (f∘g)(x)=f(g(x))=2*(x^2)+3, (g∘f)(x)=g(f(x))=(2*x+3)^2

Helping Notes

Only two inputs are required: f(x) and g(x). Evaluation at a point is optional in many tools, so we omit it.
We symbolically substitute: (f∘g)(x)=f(g(x)) and (g∘f)(x)=g(f(x)), then simplify.
Watch domains if you later plug numbers in (e.g., log(x) requires x>0).

Results

(f∘g)(x) = f(g(x))

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(g∘f)(x) = g(f(x))

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Simplified Forms

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What is a Composite Function Calculator?

A Composite Function Calculator constructs and simplifies function compositions such as \((f\circ g)(x)=f(g(x))\) and \((g\circ f)(x)=g(f(x))\). Composition lets you apply one function’s output as another function’s input, enabling multi-stage models: unit conversions, nested transformations, and chained operations in algebra, calculus, and data pipelines. The calculator returns a simplified expression, evaluates numerically at specified \(x\)-values, and emphasizes the domain of the composite, which may be stricter than the individual domains. It can also compare \((f\circ g)\) vs. \((g\circ f)\) to highlight that order matters, and—when enabled—checks inverse relationships \((f\circ f^{-1})(x)=x\) on valid domains. Clear, step-by-step working helps students follow substitution, distribute exponents, rationalize radicals, and combine like terms.

About the Composite Function Calculator

The tool parses \(f(x)\) and \(g(x)\), substitutes carefully, and simplifies using algebraic identities (powers, factoring, trig identities) while tracking restrictions such as denominators \(\neq0\), even roots requiring nonnegative radicands, and logarithms needing positive arguments. It supports symbolic constants and parameters, detects removable vs. essential discontinuities, and can produce piecewise composites when inputs are piecewise-defined. To support calculus workflows, it optionally previews the derivative of the composite via the chain rule and provides numerical checks at sample points. If an inverse function is supplied, the calculator verifies compositions reduce to the identity on the intersection of their domains.

How to Use this Composite Function Calculator

Enter \(f(x)\) and \(g(x)\) in standard notation (use parentheses generously to avoid ambiguity).
Choose which composite to build: \((f\circ g)(x)=f(g(x))\) or \((g\circ f)(x)=g(f(x))\).
Click Calculate to get the simplified expression, domain restrictions, and (optionally) a derivative preview via the chain rule.
Evaluate the composite at example \(x\)-values to confirm behavior and check for undefined points.
Copy the simplified result and equation preview for assignments, lab reports, or code.

Core Formulas (LaTeX)

Definition of composition: \[ (f\circ g)(x)=f(g(x)),\qquad (g\circ f)(x)=g(f(x)). \]

Domain of a composite: \[ \operatorname{Dom}(f\circ g)=\{\,x\in\operatorname{Dom}(g)\mid g(x)\in\operatorname{Dom}(f)\,\}. \]

Chain rule (derivative preview): \[ \frac{d}{dx}\big[f(g(x))\big]=f'\!\big(g(x)\big)\cdot g'(x). \]

Inverse check (when exists): \[ (f\circ f^{-1})(x)=x \ \text{on }\operatorname{Dom}(f),\qquad (f^{-1}\circ f)(x)=x \ \text{on }\operatorname{Dom}(f^{-1}). \]

Examples (Illustrative)

Example 1 — Polynomial with linear inside

\(f(x)=x^{2}\), \(g(x)=2x+3\). Then \((f\circ g)(x)=f(g(x))=(2x+3)^2=4x^2+12x+9\). \((g\circ f)(x)=2x^2+3\) (different expression; order matters).

Example 2 — Radical domain restriction

\(f(x)=\sqrt{x}\), \(g(x)=x-1\). \((f\circ g)(x)=\sqrt{x-1}\) with domain \(x\ge1\). \((g\circ f)(x)=\sqrt{x}-1\) with domain \(x\ge0\).

Example 3 — Trig and logarithm

\(f(x)=\ln x\) (domain \(x>0\)), \(g(x)=\sin x\). \((f\circ g)(x)=\ln(\sin x)\) defined when \(\sin x>0\). \((g\circ f)(x)=\sin(\ln x)\) defined when \(x>0\).