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4MA1/2HR Jun 2019, Question 18: Inverse Functions

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 2HR (Higher) · Calculator · 9 marks

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The question, in short

Two functions, one a fraction in x and one a shift: the value excluded from the domain, a composite, an inverse, and a gradient estimated from a drawn curve.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (a) · 1 mark

Value of x that makes the denominator zero.

  1. 1f(x) = x/(4x − 3), and a fraction cannot have 0 on the bottom
  2. 24x − 3 = 0 when x = 3/4, so that value must be excluded

Answer3/4

Part (b) · 2 marks

Put g(x) into f and simplify.

  1. 1fg(x) = f(x − 5)
  2. 2f(x − 5) = (x − 5)/(4(x − 5) − 3)
  3. 34(x − 5) − 3 = 4x − 20 − 3 = 4x − 23
  4. 4So fg(x) = (x − 5)/(4x − 23)

Answer(x − 5)/(4x − 23)

Part (c) · 3 marks

Find the inverse of f.

  1. 1Write y = x/(4x − 3) and swap the letters: x = y/(4y − 3)
  2. 2Multiply by (4y − 3): x(4y − 3) = y
  3. 3Expand: 4xy − 3x = y
  4. 4Collect the y terms: 4xy − y = 3x, so y(4x − 1) = 3x
  5. 5f⁻¹(x) = 3x/(4x − 1)

Answer3x/(4x − 1)

Part (d) · 3 marks

Draw a tangent at x = −0.5 and work out its gradient.

  1. 1Draw a tangent to the curve at x = −0.5
  2. 2Pick two points on the tangent as far apart as it is easy to read
  3. 3Gradient = change in y ÷ change in x for those two points
  4. 4A tangent drawn accurately gives a gradient of about 6
  5. 5Any value from 5 to 7 is accepted

Answer5 to 7

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