Function Composition Calculator — Evaluate f(g(x))

Enter two function rules and a value of x to evaluate the inner function first, then the outer function. Your expressions stay in your browser and are never uploaded or stored.

Get the composed rule, the value of g(x), the final value of f(g(x)), substitution steps, and an order comparison with g(f(x)).

Define the functions and input

Enter a finite number between −10¹² and 10¹². Scientific notation such as 1e-3 is accepted.

f(g(x)) = 2 · (x ^ 2 − 1) + 3Evaluate at x = 4

Use x, numbers, + − * / ^, and parentheses. Implicit multiplication such as 2x is accepted.

Supported functions and constants

Use sqrt, abs, exp, ln, log (base 10), sin, cos, tan, asin, acos, or atan, plus constants pi and e. Trigonometric angles are in radians. Enter only the right-hand side of each function rule.

Try an example

Composition result

Final value
f(g(4)) = 33
Evaluate g first, then use its output as the input of f.
Composed rulef(g(x)) = 2 · (x ^ 2 − 1) + 3
Inner valueg(4) = 15
Outer substitutionf(15) = 2 · 15 + 3
Reverse orderg(f(4)) = 120

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Step-by-step composition

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How function composition works

1. Identify the order

In f(g(x)), g is inside the parentheses, so evaluate g first. The result becomes the input of f.

2. Substitute the rule

To write a composed rule, replace every x in f(x) with the entire expression g(x). Parentheses preserve the intended operations.

3. Check both domains

The original input must be allowed by g, and g(x) must be an allowed input for f. Either condition can make the composition undefined.

4. Keep the order

Composition is generally not commutative: f(g(x)) and g(f(x)) often have different rules and values.

Definition

(f ∘ g)(x) = f(g(x))

For f(x)=2x+3 and g(x)=x²−1, substitution gives f(g(x))=2(x²−1)+3. At x=4, first find g(4)=15, then f(15)=33.

Composition examples and domain checks

Linear after quadratic

If f(x)=3x−2 and g(x)=x²+1, then f(g(x))=3(x²+1)−2=3x²+1.

Radical outer function

If f(x)=sqrt(x), then the inner output must be nonnegative. For g(x)=x−5, the real composition requires x≥5.

Reciprocal inner function

If g(x)=1/x, then x=0 is excluded before the outer function is evaluated.

Order changes the result

With f(x)=x+1 and g(x)=x², f(g(x))=x²+1, while g(f(x))=(x+1)².

Accuracy, privacy, and limits

The calculator parses a restricted mathematical grammar and evaluates it locally; it does not use JavaScript eval and does not send function rules or values to a server. Intermediate and final calculations use browser floating-point arithmetic and display up to 12 significant digits.

Inputs are limited to 180 characters, 240 expression nodes, 40 levels of nesting, and |x|≤10¹². Undefined real operations—such as division by zero, sqrt(−1), ln(0), or a non-real power—produce an error instead of a misleading result. The composed expression is substitution form, not a full symbolic simplification. Last reviewed: 4 August 2026.

Function composition FAQs

What does f(g(x)) mean?

It means apply g to x first, then apply f to the output. It is also written (f∘g)(x).

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

Usually not. Reversing the order changes which function receives the original input and can change both the value and the allowed domain.

How do I create the composed rule?

Replace every occurrence of x in the outer rule f(x) with the full inner rule g(x). Put the inner rule in parentheses when needed.

Why is my composition undefined?

The input may cause division by zero or an invalid logarithm or root in g. Alternatively, g may return a value outside the domain of f. The error message identifies which stage failed.

Does log mean base 10 or base e?

log(x) means base 10, while ln(x) means the natural logarithm. Both require a positive real input.

Are trigonometric inputs in degrees?

No. The calculator uses radians, matching the standard JavaScript mathematical functions. For example, enter pi/2 for 90 degrees.

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