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Cambridge IGCSE0580

Functions

Mathematics 0580 Chapter Notes

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Functions
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Functions notes

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In this chapter

  1. 1Function notation
  2. 2Domain and range
  3. 3Finding an inverse function
  4. 4What an inverse does
  5. 5Composite functions
  6. 6A composite with a squared bracket

1. Function notation

f(x) = 3x - 5 is written in function notation. It means: take x, multiply by 3, subtract 5.

f(4) means replace x by 4. Put brackets round the replacement, so a negative number is squared correctly.

To find x when f(x) has a given value, form an equation and solve it. Do not substitute the value for x.

f(a) = the output when x = a

Examiner insight

“The most common error involved substitution of x = 15 leading to an answer of 61.”

Worked example 12 marks

f(x) = 3x + 1 g(x) = 5 - 3x

(i) Find x when f(x) = g(x).

  1. 1

    3x + 1 = 5 - 3x [1]

  2. 2

    6x = 4, so x = 2/3 [1]

2. Domain and range

The domain is the set of input values. The range is the set of output values.

For a domain given as a list, work out f(x) for each value. The range is the list of outputs.

The range is not the statistical range, so do not subtract. Do not fill in the values between the outputs.

Examiner insight

“Some candidates misunderstood the concept of range and treated it as a statistical range, giving 10 (from 14 − 4) as the answer.” “Some candidates gave a list of every integer from 4 to 14 as their answer or gave an answer as an interval 4 ≤ x ≤ 14.”

Worked example 12 marks

f(x) = 10 - 2x

(i) The domain of f(x) is {-2, 0, 3}. Find the range of f(x).

  1. 1

    f(-2) = 10 - 2 × (-2) = 14 [1]

  2. 2

    f(0) = 10 and f(3) = 10 - 6 = 4, so the range is {14, 10, 4} [1]

3. Finding an inverse function

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

To find it, write y = f(x). Swap x and y, then make y the subject. Give the answer in terms of x.

f⁻¹(x) is not 1/(f(x)). The -1 does not mean a reciprocal.

x = f(y) ⇒ y = f⁻¹(x)

Examiner insight

“Common errors were not dividing every term by 2 or sign errors when moving terms across the equal sign or forgetting to switch x and y.” “Some candidates found the reciprocal of the function rather than the inverse function.”

Worked example 12 marks

h(x) = 7 + 2x

(b) Find h⁻¹(x).

  1. 1

    x = 7 + 2y [1]

  2. 2

    2y = x - 7, so h⁻¹(x) = (x - 7)/2 [1]

4. What an inverse does

Applying f and then f⁻¹ returns the starting value, so f⁻¹f(x) = x. The inverse returns the starting value.

If h⁻¹(a) = b, then h(b) = a. To solve h⁻¹(x) = k, find h(k).

To find h⁻¹(k) instead, solve h(x) = k. Do not calculate h(k).

f⁻¹f(x) = x

h⁻¹(x) = k ⇒ x = h(k)

Examiner insight

“Most stronger candidates were able give their answer with no working, realising that applying the inverse function to a function got you back to the starting point.” “Many made the common error of calculating h(10 000).”

Worked example 11 mark

h(x) = 10^(2/x), x ≠ 0

(b) Find h⁻¹(10 000).

  1. 1

    Solve h(x) = 10 000: 10^(2/x) = 10⁴

  2. 2

    2/x = 4, so x = 0.5 [1]

5. Composite functions

gf(x) means g(f(x)). Do f first, then g. The function nearest x acts first.

To form it, replace x in g by the whole of f(x). It is not a product g(x) × f(x).

gg(x) means apply g twice. For a number, work in two stages: find the inner value first.

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

Examiner insight

“The gf(x) was most commonly correct with a common error in fg(x + 1) being to forget the +1.”

Worked example 12 marks

f(x) = 3x + 1 g(x) = 5 - 3x

(iii) Find gg(x). Give your answer in its simplest form.

  1. 1

    gg(x) = 5 - 3(5 - 3x) [1]

  2. 2

    = 5 - 15 + 9x = 9x - 10 [1]

6. A composite with a squared bracket

When f(x) is put into a squared term, square the whole bracket. Expand it as a double bracket.

(a + b)² = a² + 2ab + b². The middle term is not missing.

Collect like terms and give the answer in the form asked.

(a - b)² = a² - 2ab + b²

Examiner insight

“Many candidates made sign errors when expanding the squared bracket, notably with the squared term.” “Some candidates squared the two terms in the bracket rather than expanding double brackets.”

Worked example 13 marks

f(x) = 10 - 2x g(x) = 2x² + 3

(iii) Find gf(x) in the form ax² + bx + c.

  1. 1

    gf(x) = 2(10 - 2x)² + 3 [1]

  2. 2

    (10 - 2x)² = 100 - 40x + 4x² [1]

  3. 3

    gf(x) = 200 - 80x + 8x² + 3 = 8x² - 80x + 203 [1]

Recap

  • Extended: f(a) is the output when x = a. Use brackets when substituting.
  • Extended: to find x when f(x) = k, form and solve an equation.
  • Extended: the domain is the set of inputs. The range is the set of outputs.
  • Extended: for f⁻¹(x), swap x and y, then make y the subject.
  • Extended: f⁻¹f(x) = x. h⁻¹(x) = k means x = h(k).
  • Extended: gf(x) = g(f(x)). Do f first.
  • Extended: put the whole of f(x) in place of x, and expand squared brackets in full.

End-of-chapter exercise

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

  1. Do not use a calculator. f(x) = 4x - 7. Find f(-3).2 marks
  2. Do not use a calculator. g(x) = x² - 5. Find g(-4).2 marks
  3. f(x) = 3x + 8. Find x when f(x) = -1.2 marks
  4. Do not use a calculator. f(x) = 2x² - 3. The domain of f(x) is {-2, 0, 1, 3}. Find the range of f(x).3 marks
  5. f(x) = 5x - 2. Find f⁻¹(x).2 marks
  6. g(x) = (3(x + 4))/5. Find g⁻¹(x).3 marks
  7. Do not use a calculator. h(x) = 3^(x). Find x when h⁻¹(x) = 4.2 marks
  8. f(x) = 2/(x + 3) and g(x) = (2x + 1)². Find fg(x). Give your answer as a fraction in its simplest form.3 marks
  9. Do not use a calculator. f(x) = 2x - 3 and g(x) = x² + 1. (a) Find gf(x) in the form ax² + bx + c. (b) Solve gf(x) = 4g(x).5 marks
  10. A scientist grows cells in a dish. After x hours the number of cells, in thousands, is h(x) = 5 × 2^(x). When there are x thousand cells, the mass of the cells, in milligrams, is m(x) = 3x + 4. (a) Find mh(x), the mass of the cells after x hours. (b) Find h⁻¹(40). (c) Find the number of hours after which the mass of the cells is 244 mg.7 marks

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