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Cambridge IGCSE0607

Functions

International Mathematics 0607 Chapter Notes

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Functions
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In this chapter

  1. 1Function notation
  2. 2Mapping diagrams, domain and range
  3. 3Finding an inverse function
  4. 4What an inverse means
  5. 5Forming a composite function
  6. 6Composite functions with numbers and equations

1. Function notation

A function is a rule that turns an input into an output. f(x) is the output of the function f when the input is x. This is function notation.

To find f(3), replace every x in the rule by 3 and work it out. Put a negative input in brackets.

To find x when f(x) is given, make the rule equal to that value and solve for x.

f(x) = 3x - 5

f(-2) = 3 × (-2) - 5 = -11

Worked example 15 marks

f(x) = (5(x-2))/4

(a) Find f(3).

(b) Find the value of x when f(x) = 20.

  1. 1

    (a) f(3) = (5(3-2))/4 [1]

  2. 2

    = 5/4 [1]

  3. 3

    (b) (5(x-2))/4 = 20 [1]

  4. 4

    5(x-2) = 80, so x - 2 = 16 [1]

  5. 5

    x = 18 [1]

2. Mapping diagrams, domain and range

A mapping diagram lists inputs on one side and the outputs they map to on the other. The inputs are the domain and the outputs are the range.

To complete a mapping diagram, link the pattern in the x-values with the pattern in the f(x)-values. Write the rule as f(x), then use it both ways.

A rule of the form f(x) = ax + b has a as the change in f(x) for each 1 added to x.

f(x) = ax + b

Examiner insight

“Some saw a pattern of ‘+ 4’ in the first three values in the range so gave 13 + 4 = 17 when x = 7. This pattern is only correct when the x-values increase by 1 each time.”

Worked example 12 marks

Complete the mapping diagram.

xf(x)
15
29
313
7?
?41
  1. 1

    x goes up by 1 and f(x) goes up by 4, so f(x) = 4x + 1. Then f(7) = 4 × 7 + 1 = 29 [1]

  2. 2

    4x + 1 = 41, so 4x = 40 and x = 10 [1]

3. Finding an inverse function

The inverse function f⁻¹(x) undoes f. It takes an output of f back to the input.

Write y = f(x). Interchange x and y, then rearrange to make y the subject. The answer must be in terms of x.

f⁻¹(x) is not the reciprocal 1/(f(x)).

y = 2x - 3 → x = 2y - 3 → y = (x+3)/2

Examiner insight

“Some candidates correctly rearranged the expression and forgot to swap the variables.” “A small number of candidates confused inverse with reciprocal.”

Worked example 12 marks

f(x) = 2x - 3

Find f⁻¹(x).

  1. 1

    y = 2x - 3. Interchange x and y: x = 2y - 3, so 2y = x + 3 [1]

  2. 2

    f⁻¹(x) = (x+3)/2 [1]

4. What an inverse means

f⁻¹(4) = x means f(x) = 4. To find f⁻¹ of a number, solve f(x) equal to that number. There is no need to find f⁻¹(x) first.

An inverse undoes its function, so ff⁻¹(x) = x.

f⁻¹(a) = x iff f(x) = a

ff⁻¹(x) = x

Examiner insight

“The stronger candidates realised that if f –1(4) = x then f(x) = 4 and this led to the very simple equation 3x – 2 = 4 to solve.” “A few strong candidates understood that f(f –1(x)) = x as long as a function has an inverse.”

Worked example 12 marks

f(x) = 3x - 2

Find f⁻¹(7).

  1. 1

    f⁻¹(7) = x means f(x) = 7, so 3x - 2 = 7 [1]

  2. 2

    3x = 9, so f⁻¹(7) = 3 [1]

5. Forming a composite function

A composite function is one function applied after another. gf(x) means g(f(x)): put f(x) into g.

Work from the inside. Write f(x) in brackets in place of x in g, then expand and simplify. Where the answer is a fraction, give it in its simplest form.

gf(x) is not g(x) × f(x). The order matters: in gf(x), f is applied first.

gf(x) = g(f(x))

Examiner insight

“Some candidates who expanded the brackets correctly forgot to add on the 1 to their answer.”

Worked example 13 marks

f(x) = 2x - 3 and g(x) = x² + 1

Find gf(x). Give your answer in the form ax² + bx + c.

  1. 1

    gf(x) = g(2x - 3) [1]

  2. 2

    = (2x - 3)² + 1 [1]

  3. 3

    = 4x² - 12x + 9 + 1 = 4x² - 12x + 10 [1]

6. Composite functions with numbers and equations

For a number, find the inside value first. fg(2) means find g(2), then put the result into f.

To solve an equation such as ff(x) = 46, form the composite function, set it equal to the number and solve.

When a composite function is given, such as fh(x) = 8x - 1 with h(x) = px + q, substitute h(x) into f and equate coefficients.

fg(2) = f(g(2))

Examiner insight

“Finding g(1) and then substituting this value into f(x) should be seen as a much simpler approach.” “Nearly all candidates showed a good understanding of functions with most candidates finding g(2) first before substituting their answer into f(x).”

Worked example 13 marks

f(x) = 3x - 2

Solve ff(x) = 46.

  1. 1

    ff(x) = f(3x - 2) = 3(3x - 2) - 2 [1]

  2. 2

    = 9x - 8, so 9x - 8 = 46 [1]

  3. 3

    9x = 54, so x = 6 [1]

Recap

  • f(x) is the output for input x: replace x by the input, in brackets.
  • To solve f(x) = k, set the rule equal to k and solve for x.
  • Domain is the set of inputs and range is the set of outputs.
  • Inverse function: interchange x and y, then make y the subject.
  • f⁻¹(a) = x means f(x) = a. Also ff⁻¹(x) = x.
  • gf(x) = g(f(x)): f is applied first.
  • Composite with a number: work out the inside function first.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. f(x) = 3x - 5. Find f(-2).1 mark
  2. Do not use a calculator. h(x) = 2x² + 3. Find h(-3).2 marks
  3. Do not use a calculator. g(x) = (3(x+4))/5. Find the value of x when g(x) = 6.3 marks
  4. Do not use a calculator. A linear function f maps 1 to 7, 2 to 10 and 4 to 16. Find f(7).3 marks
  5. Do not use a calculator. f(x) = 5x + 2. Find f⁻¹(x).2 marks
  6. f(x) = 2x - 7. Find f⁻¹(5).2 marks
  7. f(x) = 3x - 1 and g(x) = x² - 5. Find gf(x). Give your answer in the form ax² + bx + c.3 marks
  8. f(x) = 4/(x+1) and g(x) = (2x+3)². Find fg(x). Give your answer as a fraction in its simplest form.3 marks
  9. Do not use a calculator. f(x) = 3x - 4 and h(x) = px + q. It is given that hf(x) = 12x - 11. (a) Find the value of p and the value of q. (b) Solve h⁻¹(x) = 2.5 marks
  10. A workshop prices a job in two stages. A job lasting x hours has a cost of f(x) = 6x + 20 dollars. A cost of x dollars is then turned into the customer's bill of g(x) = (5x - 10)/4 dollars. (a) Show that the bill for a job lasting x hours is (15x + 45)/2 dollars. (b) A customer's bill is 105 dollars. Find how many hours the job lasted. (c) Find g⁻¹(x). Work out g⁻¹(105) and state what it tells you about this job.7 marks

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