Inverse and Composite Functions
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Revision notes for CIE IGCSE Maths Inverse and Composite Functions. Open the guide for explanations and worked examples. Written against the CIE IGCSE Maths (0580) specification, so the content matches what's examinable rather than general Maths background.

Inverse and Composite Functions

What you'll learn

  • How to use function notation, including inputs, outputs, domains and ranges.
  • How to find inverse functions by rearranging.
  • Why some functions, especially quadratics, need restricted domains.
  • How to calculate composite functions such as fg(2)fg(2)fg(2) and f(g(x))f(g(x))f(g(x)).

1. Function notation: inputs and outputs

A function is like a rule machine: you put in an input, the rule does something, and you get an output.

For example, if f(x)=3x−2f(x)=3x-2f(x)=3x−2, then f(5)f(5)f(5) means “put 5 into the function fff”.

Definition

Function, domain and range

  • A function is a rule that gives exactly one output for each allowed input.
  • The domain is the set of allowed input values.
  • The range is the set of possible output values.

Worked example: evaluating a function

Example

Finding a function value

A function is defined by h(x)=2x−75h(x)=\frac{2x-7}{5}h(x)=52x−7​. Find h(−4)h(-4)h(−4).

  1. Substitute −4-4−4 wherever you see xxx:

    h(−4)=2(−4)−75h(-4)=\frac{2(-4)-7}{5}h(−4)=52(−4)−7​

  2. Simplify the numerator:

    h(−4)=−8−75h(-4)=\frac{-8-7}{5}h(−4)=5−8−7​

  3. Divide by 5:

    h(−4)=−155=−3h(-4)=\frac{-15}{5}=-3h(−4)=5−15​=−3

  4. So the answer is h(−4)=−3h(-4)=-3h(−4)=−3.

Tip

Substitution check

If the input is negative, use brackets when substituting. For example, write 2(−4)2(-4)2(−4), not just 2−42-42−4.

2. Domains and excluded values

Sometimes not every value of xxx is allowed.

Two common restrictions are:

  • You cannot divide by zero.
  • You cannot square root a negative number if you are working with real numbers.

Worked example: values that cannot be included

Example

Finding excluded values

A function is defined by p(x)=6x−3p(x)=\frac{6}{x-3}p(x)=x−36​.

  1. Look at the denominator:

    x−3x-3x−3

  2. The denominator must not equal zero:

    x−3≠0x-3 \neq 0x−3=0

  3. Solve the excluded-value equation:

    x≠3x \neq 3x=3

  4. So 3 cannot be included in any domain of ppp.

For square roots, look at what is inside the square root.

Example

Square-root domain restriction

A function is defined by q(x)=15−xq(x)=\sqrt{15-x}q(x)=15−x​. Which values cannot be included in any domain of qqq?

  1. The expression inside the square root must be non-negative:

    15−x≥015-x \ge 015−x≥0

  2. Solve the inequality:

    x≤15x \le 15x≤15

  3. Therefore values greater than 15 cannot be included in any domain of qqq.

3. Inverse functions

An inverse function reverses the effect of the original function.

If fff sends 2 to 7, then f−1f^{-1}f−1 sends 7 back to 2.

Definition

Inverse function

The inverse function of fff, written f−1f^{-1}f−1, is the function that undoes fff. The domain of fff becomes the range of f−1f^{-1}f−1, and the range of fff becomes the domain of f−1f^{-1}f−1.

On a graph, a function and its inverse are reflections in the line y=xy=xy=x. This is because the input and output coordinates swap places.

Diagram showing a function and its inverse reflected in y=x, with domain and range swapping

Common Mistake

Inverse does not mean reciprocal

f−1(x)f^{-1}(x)f−1(x) does not mean 1f(x)\frac{1}{f(x)}f(x)1​. It means the function that reverses fff.

How to find an inverse function

For most algebraic inverses:

  1. Write y=f(x)y=f(x)y=f(x).
  2. Swap xxx and yyy.
  3. Rearrange to make yyy the subject.
  4. Replace yyy with f−1(x)f^{-1}(x)f−1(x).

Worked example: inverse of a linear function

Example

Finding a linear inverse

Let f(x)=4x−73f(x)=\frac{4x-7}{3}f(x)=34x−7​. Find f−1(x)f^{-1}(x)f−1(x).

  1. Start by writing the function as y=y=y=:

    y=4x−73y=\frac{4x-7}{3}y=34x−7​

  2. Swap xxx and yyy:

    x=4y−73x=\frac{4y-7}{3}x=34y−7​

  3. Rearrange to make yyy the subject. First multiply by 3:

    3x=4y−73x=4y-73x=4y−7

  4. Add 7 to both sides:

    3x+7=4y3x+7=4y3x+7=4y

  5. Divide by 4:

    y=3x+74y=\frac{3x+7}{4}y=43x+7​

  6. Write the inverse function:

    f−1(x)=3x+74f^{-1}(x)=\frac{3x+7}{4}f−1(x)=43x+7​

4. Quadratic inverses and completing the square

A quadratic function does not always have an inverse over all real values, because two different inputs can give the same output.

For example, x=2x=2x=2 and x=−2x=-2x=−2 both give x2=4x^2=4x2=4.

Definition

One-to-one function

A function is one-to-one if each output comes from exactly one input. A function needs to be one-to-one on its domain for its inverse to be a function.

This is why exam questions often give a restricted domain such as x≥−3x \ge -3x≥−3. That tells you which branch of the quadratic to use.

Common Mistake

Quadratics need a chosen branch

When finding the inverse of a quadratic, the domain restriction decides whether you take the positive or negative square-root branch.

Completing the square recap

A quadratic such as x2+10x+7x^2+10x+7x2+10x+7 can be rewritten by completing the square:

x2+10x+7=(x+5)2−18x^2+10x+7=(x+5)^2-18x2+10x+7=(x+5)2−18

This form is useful because it shows the turning point and makes the inverse easier to find.

Worked example: inverse of a quadratic with a restricted domain

Example

Finding a quadratic inverse

The function ggg has domain x≥−5x \ge -5x≥−5 and is defined by g(x)=x2+10x+7g(x)=x^2+10x+7g(x)=x2+10x+7.

Find the range of g−1g^{-1}g−1 and find g−1(x)g^{-1}(x)g−1(x).

  1. The range of g−1g^{-1}g−1 is the same as the domain of ggg. Since the domain of ggg is x≥−5x \ge -5x≥−5, the outputs of g−1g^{-1}g−1 are all values at least -5.

  2. Complete the square:

    g(x)=(x+5)2−18g(x)=(x+5)^2-18g(x)=(x+5)2−18

  3. Write y=(x+5)2−18y=(x+5)^2-18y=(x+5)2−18:

    y=(x+5)2−18y=(x+5)^2-18y=(x+5)2−18

  4. Rearrange to isolate the square:

    y+18=(x+5)2y+18=(x+5)^2y+18=(x+5)2

  5. Take the square root. Since the original domain is x≥−5x \ge -5x≥−5, we know x+5≥0x+5 \ge 0x+5≥0, so we choose the positive square root:

    y+18=x+5\sqrt{y+18}=x+5y+18​=x+5

  6. Make xxx the subject:

    x=−5+y+18x=-5+\sqrt{y+18}x=−5+y+18​

  7. Swap the letter back so the inverse is written as a function of xxx:

    g−1(x)=−5+x+18g^{-1}(x)=-5+\sqrt{x+18}g−1(x)=−5+x+18​
Common Mistake

Forgetting the domain restriction

Do not automatically write ±x\pm\sqrt{x}±x​. The given domain tells you which sign to use. If the domain is to the right of the turning point, you usually take the positive square-root branch.

5. Composite functions

A composite function means doing one function after another.

Definition

Composite function

The composite function fg(x)fg(x)fg(x) means f(g(x))f(g(x))f(g(x)). You apply ggg first, then apply fff to the result.

This order is important: fg(x)fg(x)fg(x) usually gives a different answer from gf(x)gf(x)gf(x).

Key Idea

Read composites from right to left

In fg(2)fg(2)fg(2), the input 2 goes into ggg first. Then the answer from ggg goes into fff.

Worked example: calculating a composite value

Example

Finding fg(1)

Let f(x)=3x−2f(x)=\frac{3}{x-2}f(x)=x−23​ and g(x)=x+5g(x)=x+5g(x)=x+5. Find fg(1)fg(1)fg(1).

  1. Rewrite the notation:

    fg(1)=f(g(1))fg(1)=f(g(1))fg(1)=f(g(1))

  2. Find g(1)g(1)g(1) first:

    g(1)=1+5=6g(1)=1+5=6g(1)=1+5=6

  3. Now put 6 into fff:

    f(6)=36−2f(6)=\frac{3}{6-2}f(6)=6−23​

  4. Simplify:

    f(6)=34f(6)=\frac{3}{4}f(6)=43​

  5. Therefore fg(1)=34fg(1)=\frac{3}{4}fg(1)=43​.

Worked example: composite involving a square root

Example

Using the inner function first

Let f(x)=5x−12f(x)=\frac{5x-1}{2}f(x)=25x−1​ and g(x)=17−xg(x)=\sqrt{17-x}g(x)=17−x​. Find fg(8)fg(8)fg(8).

  1. The notation means:

    fg(8)=f(g(8))fg(8)=f(g(8))fg(8)=f(g(8))

  2. Work out g(8)g(8)g(8):

    g(8)=17−8g(8)=\sqrt{17-8}g(8)=17−8​

  3. Simplify the square root:

    g(8)=9=3g(8)=\sqrt{9}=3g(8)=9​=3

  4. Now put 3 into fff:

    f(3)=5(3)−12f(3)=\frac{5(3)-1}{2}f(3)=25(3)−1​

  5. Simplify:

    f(3)=142=7f(3)=\frac{14}{2}=7f(3)=214​=7

  6. So fg(8)=7fg(8)=7fg(8)=7.

Exam technique

In the exam

  1. For an inverse, write y=y=y= first, swap xxx and yyy, then rearrange carefully.
  2. For a quadratic inverse, complete the square and use the domain restriction to choose the correct square-root sign.
  3. For a composite function, work from right to left: in fg(a)fg(a)fg(a), find g(a)g(a)g(a) before using fff.
Self review

Check yourself

  • If f(x)=5x−4f(x)=5x-4f(x)=5x−4, can you find f−1(x)f^{-1}(x)f−1(x)?
  • Which value must be excluded from the domain of h(x)=7x+2h(x)=\frac{7}{x+2}h(x)=x+27​?
  • Why does p(x)=x2p(x)=x^2p(x)=x2 need a restricted domain before it can have an inverse function?

Recap questions

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