PrepWithTee owl logo PrepWithTee
Functions

Functions

Composite functions

Type 3 Composite Functions Extended 5 min read

On this page
  1. In this note
  2. Formula bank
  3. One function after another
  4. Ways they test it
  5. Mistakes to avoid
  6. Summary
  7. Practice
  8. Can I do this?
  9. Quiz

In this note

Syllabus 2.12/2.13: composite functions Extended
  • 2.12Understand a composite function such as \(fg(x)\): do \(g\) FIRST, then \(f\).
  • 2.12Find \(fg(3)\), \(gf(x)\), \(ff(x)\) and simplify.
  • 2.12Solve equations such as \(gf(x)=13\).

Before you start: Function notation. Next: Domain and range.

Formula bank

CompositeLearn it\(fg(x)=f\big(g(x)\big)\)the function NEXT TO \(x\) acts first.Use it when two function letters side by side.
With a numberLearn it\(fg(3)\): work out \(g(3)\), then put the answer into \(f\)Use it when a number in the brackets.

One function after another

\(fg(x)\) means: put \(x\) into \(g\), then put the answer into \(f\). The function nearest to x acts first - read it right to left. Usually \(fg(x)\ne gf(x)\).

With \(f(x)=2x+1\) and \(g(x)=x^{2}\): \(gf(3)=g(7)=49\), but \(fg(3)=f(9)=19\).

General methodHow to find a composite function

  1. With a number: work out the inside function first, then put that answer into the outside function.
  2. With x: take the OUTSIDE function's rule and replace its x by the whole inside rule, in brackets.
  3. Expand and simplify if asked (e.g. 'in the form ax² + bx + c').
  4. To solve fg(x) = k: write fg(x) out, then solve the equation.

Example \(f(x)=2x+1\), \(g(x)=x^{2}\). Find \(gf(x)\).

\(f\) acts first: put \((2x+1)\) into \(g\).

  1. \(g(2x+1)=(2x+1)^{2}\)
  2. \(=4x^{2}+4x+1\)

Answer \(4x^{2}+4x+1\)

Try it yourself

1
\(f(x)=2x+1\), \(g(x)=x^{2}\). Find \(fg(3)\).
Hint
\(g(3)=9\), then \(f(9)\).
Show answer

Answer 19

2
\(f(x)=2x+1\). Find \(ff(x)\).
Show answer

Answer \(4x+3\)

3
\(f(x)=2x+1\), \(g(x)=x^{2}\). Solve \(fg(x)=19\).
Show answer

Answer 3, −3

Ways they test it

Type 1 Composite of a number

"\(f(x)=3x-1\), \(g(x)=x^{2}+2\). Find \(fg(2)\)." (2 marks)

Spot it two letters and a number.

  1. Inside first: g(2) = 6.
  2. Then f(6).

For example \(g(2)=6\)

  1. \(f(6)=17\)

Answer 17

Type 2 Composite in x

"Find \(gf(x)\) in the form \(ax^{2}+bx+c\)." (2 marks)

Spot it x inside the brackets.

  1. Outside rule with x replaced by the inside rule.
  2. Expand.

For example \(g(2x-3)=(2x-3)^{2}+4\)

    Answer \(4x^{2}-12x+13\)

    Type 3 Solve a composite

    "Solve \(gf(x)=13\)." (3 marks)

    Spot it 'solve' with a composite.

    1. Write gf(x) out.
    2. Make it = 0 and solve.

    For example \(4x^{2}-12x+13=13\)

    1. \(4x(x-3)=0\)

    Answer \(x=0\) or \(3\)

    Type 4 Fractions

    "\(p(x)=\dfrac3{x+2}\), \(q(x)=2x-1\). Find \(pq(x)\)."

    Spot it a function with a denominator.

    1. Put the inside rule into the denominator.
    2. Simplify.

    For example \(\dfrac3{(2x-1)+2}\)

      Answer \(\dfrac3{2x+1}\)

      Mistakes to avoid

      • ✗ \(fg(x)\): doing f first✓ g firstThe function next to x acts first.
      • ✗ \(fg(x)=f(x)\times g(x)\)✓ \(f\big(g(x)\big)\)Composite means one inside the other, not multiplied.
      • ✗ \(g(2x+1)=2x^{2}+1\)✓ \((2x+1)^{2}\)Replace x by the WHOLE bracket.

      Summary

      • \(fg(x)=f\big(g(x)\big)\): the one nearest x acts first.
      • Number: inside first, then outside.
      • In x: outside rule with x replaced by the inside rule (in brackets).
      • Usually \(fg\ne gf\).

      If you see …

      • "find fg(3)"g(3) first, then f of the answer.
      • "solve gf(x) = …"write gf(x) out as an expression, then solve.

      Practice

      Practice: easy, medium, hard

      Easy

      1
      \(f(x)=2x+1\), \(g(x)=x^{2}\). Find \(gf(3)\).
      Show answer

      Answer 49

      2
      \(f(x)=2x+1\), \(g(x)=x^{2}\). Find \(fg(-2)\).
      Show answer

      Answer 9

      3
      \(f(x)=2x+1\), \(g(x)=x^{2}\). Find \(fg(x)\).
      Show answer

      Answer \(2x^{2}+1\)

      Medium

      1
      \(f(x)=3x-1\), \(g(x)=x^{2}+2\). Find \(fg(2)\) and \(gf(2)\).
      Guide me

      What is it asking? two numbers

      What do we know? g(2) = 6 and f(2) = 5

      Plan f(6), g(5)

      Show working
      1. \(fg(2)=f(6)=17\)
      2. \(gf(2)=g(5)=27\)

      Answer 17 and 27

      2
      \(f(x)=2x-3\), \(g(x)=x^{2}+4\). Find \(gf(x)\) in the form \(ax^{2}+bx+c\).
      Guide me

      What is it asking? a quadratic

      What do we know? f first

      Plan \(g(2x-3)=(2x-3)^{2}+4\)

      Show working
      1. \((2x-3)^{2}+4\)
      2. \(4x^{2}-12x+9+4\)

      Answer \(4x^{2}-12x+13\)

      3
      \(h(x)=5-x\). Find \(hh(x)\).
      Guide me

      What is it asking? an expression

      What do we know? \(h(5-x)\)

      Plan \(5-(5-x)\)

      Show working
      1. \(5-(5-x)=x\)

      Answer \(x\)

      Hard

      1
      \(f(x)=2x-3\), \(g(x)=x^{2}+4\). Solve \(gf(x)=13\).
      Guide me

      What is it asking? x

      What do we know? \(gf(x)=4x^{2}-12x+13\)

      Plan set = 13, factorise

      Show working
      1. \(4x^{2}-12x=0\)
      2. \(4x(x-3)=0\)

      Answer 0, 3

      PartAnswer / workingMarksGuidance
      \(4x^{2}-12x+13=13\)M1
      \(4x(x-3)=0\)M1
      0, 3A1
      Total: 3 marks
      2
      \(p(x)=\dfrac3{x+2}\), \(q(x)=2x-1\). Solve \(pq(x)=1\).
      Guide me

      What is it asking? x

      What do we know? \(pq(x)=\dfrac3{2x+1}\)

      Plan \(3=2x+1\)

      Show working
      1. \(\dfrac3{2x+1}=1\)
      2. \(2x+1=3\)
      3. \(x=1\)

      Answer 1

      PartAnswer / workingMarksGuidance
      \(\dfrac3{2x+1}\)M1
      1A1
      Total: 2 marks
      3
      \(f(x)=\dfrac{x+1}2\). Solve \(ff(x)=2x-1\).
      Guide me

      What is it asking? x

      What do we know? \(ff(x)=\dfrac{x+3}4\)

      Plan \(\dfrac{x+3}4=2x-1\)

      Show working
      1. \(f\!\left(\tfrac{x+1}2\right)=\dfrac{\tfrac{x+1}2+1}2=\dfrac{x+3}4\)
      2. \(x+3=8x-4\)
      3. \(x=1\)

      Answer 1

      PartAnswer / workingMarksGuidance
      \(\dfrac{x+3}4\)M1
      \(x+3=8x-4\)M1
      1A1
      Total: 3 marks

      Can I do this?

      Anything unticked: revise that section first. All ticked: take the quiz.

      Quiz

      The multiple-choice and guided questions need JavaScript. The concept checks and exam practice below work without it.

      Your progress on this note is saved when JavaScript is on.