Composite Function Calculator

This composite function calculator finds f(g(x)) and g(f(x)) for two polynomial functions, expands them, and evaluates both at a value of x you choose. For example, with f(x) = 2x + 3 and g(x) = x², f(g(x)) = 2x² + 3 and g(f(x)) = 4x² + 12x + 9.

A number or fraction

Result

Enter values and press Calculate to see the result.

Formula

(f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x))

Apply g first, then f: replace every x in f with g(x).

(g∘f)(x)=g(f(x))(g \circ f)(x) = g(f(x))

The other order. In general f(g(x)) and g(f(x)) are different.

How to use this calculator

  1. Type f(x) and g(x) as polynomials, e.g. 2x + 3 and x^2.
  2. Choose a value of x to evaluate the composite functions at.
  3. Press Calculate to see f(g(x)) and g(f(x)) expanded, their values, and the substitution steps. To graph the function, open it in the graphing calculator.

Worked examples

f(g(x)) and g(f(x)) for a linear and a quadratic function

f(g(x)): replace x in 2x + 3 with x², giving 2x² + 3.

g(f(x)): square 2x + 3, giving 4x² + 12x + 9.

At x = 2: g(2) = 4, so f(g(2)) = 11; f(2) = 7, so g(f(2)) = 49.

A shift inside a quadratic

f(g(x)) = (x + 1)² − 1 = x² + 2x.

g(f(x)) = (x² − 1) + 1 = x².

At x = 3: f(g(3)) = f(4) = 15 and g(f(3)) = g(8) = 9.

A cubic composed with a linear function

f(g(x)) = (x − 2)³ = x³ − 6x² + 12x − 8.

g(f(x)) = x³ − 2.

At x = 1 both are −1: (1 − 2)³ = −1 and 1³ − 2 = −1.

Common mistakes

  • Multiplying instead of composing: f(g(x)) is not f(x)·g(x).
  • Working in the wrong order: f(g(x)) applies g first.
  • Substituting without brackets: with g(x) = x + 1, f(x) = x² gives (x + 1)², not x² + 1.
  • Assuming f(g(x)) = g(f(x)) — it usually isn’t.

Frequently asked questions

How do you find f of g of x?

Write f(x), then replace every x with the whole expression g(x) in brackets, and simplify. For f(x) = 2x + 3 and g(x) = x², f(g(x)) = 2x² + 3.

What is the domain of a composite function?

The x-values in the domain of g whose output g(x) is in the domain of f. For the polynomials this calculator takes, both domains are all real numbers, so f(g(x)) is defined everywhere.

Can I graph the function?

Yes — copy f(g(x)) into the Sumzo graphing calculator to plot it, see where it crosses the x-axis and read a table of values.

Which functions can I enter?

Polynomials up to degree 6, such as 3x^2 - x + 4 or (x + 1)^3. Fractions like 1/2 x work as coefficients.

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