Composition of Functions

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Composition of Functions: meaning, method and solved example (Algebra 2) – Maths PDF

In short: Function composition means feeding the output of one function straight into a second function, written f(g(x)), so that two steps of a process act as a single combined rule.

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What is Composition of Functions?

Function composition connects two machines on an assembly line: whatever the first machine produces becomes the raw material for the second, and the pair behaves like one bigger machine.

The composite of \(f\) and \(g\) is written \((f \circ g)(x) = f(g(x))\). Work from the inside out: evaluate \(g\) at \(x\), then apply \(f\) to that result. The input must belong to the domain of \(g\), and \(g(x)\) must belong to the domain of \(f\). In general \(f(g(x))\) and \(g(f(x))\) are different.

How to spot it

  • A problem shows two rules in a row, such as converting a price to a discounted price and then adding tax.
  • The notation f(g(x)) or (f ∘ g)(x) appears, with one function sitting inside the parentheses of another.
  • You are asked to use function composition to build a single formula that replaces a two-stage calculation.
  • A task asks you to compare f(g(x)) with g(f(x)) or to find the domain of a nested expression.

The method

  1. Identify the inner function (the one closest to \(x\)) and the outer function.
  2. Replace every \(x\) in the outer formula with the entire inner formula, inside parentheses.
  3. Expand and simplify, taking care with signs and squares.
  4. Check with a test value: evaluate the inner function first, then the outer, and compare with your formula.
x = -2g(x)x^2 + 37f(x)5 – 2x-9f(g(x)) = 5 – 2(x^2 + 3) = -2x^2 – 1
Input -2 goes through g to give 7, then through f to give -9.

Composition of Functions: solved example

Problem

Let \(f(x) = 5 – 2x\) and \(g(x) = x^2 + 3\). Find \(f(g(x))\), then \(g(f(x))\), and evaluate \(f(g(-2))\).

Solution

For \(f(g(x))\), put \(g(x)\) where \(x\) sits in \(f\): \(5 – 2(x^2 + 3) = -2x^2 – 1\). For \(g(f(x))\), put \(f(x)\) into \(g\): \((5 – 2x)^2 + 3 = 4x^2 – 20x + 28\).

At \(x = -2\): \(g(-2) = 7\) and \(f(7) = 5 – 14 = -9\), which agrees with \(-2(4) – 1 = -9\). So \(f(g(x)) = -2x^2 – 1\), \(g(f(x)) = 4x^2 – 20x + 28\), and \(f(g(-2)) = -9\).

Try it

Question 1

If \(h(x) = \sqrt{x}\) and \(k(x) = x – 9\), write \(h(k(x))\) and state its domain.

Show the answer

\(h(k(x)) = \sqrt{x – 9}\). The radicand must be nonnegative, so the domain is \(x \ge 9\).

Question 2

For \(f(x) = x + 6\) and \(g(x) = 3x\), compare \(f(g(1))\) with \(g(f(1))\).

Show the answer

\(f(g(1)) = f(3) = 9\), but \(g(f(1)) = g(7) = 21\). The order changes the result.

Composition of Functions vs. Transformations of Functions

Composition plugs one function into a different one. A transformation, by contrast, tweaks a single function with shifts, stretches or flips, such as \(f(x – 3) + 2\). Replacing \(x\) with \(x – 3\) is a horizontal shift, while replacing it with \(g(x)\) builds a brand-new combined rule.

Transformations of Functions

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