Unit 8

9 marks · Normal

This question tests your skill in converting metric units and finding the inverse of functions.

(a)

A field has an area of 24000 cm². Convert this area to m².

[2 marks]

(b)

Let , where is a positive real number.

i) Calculate .

(ii)

Find the inverse of , , expressing your answer in terms of .

[3 marks]

(c)

A patch of land has an area of 27 m². Using your answer from part (b), find the value of such that .

[4 marks]

Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.

Show answer

(a) 2.4 m²

(b(i)) 28

(b(ii)) f^{-1}(y) = \sqrt{y-3}

Common mistakes: Incorrect cm² to m² conversion; Algebra error in inverting function

Show mark scheme and worked solution

Marking points

M1 (a) · 1 mark · Sets up or applies correct conversion (divides by 10,000)

A1 (a) · 1 mark · Gives 2.4 m² (follow-through)

M2 (b(ii)) · 1 mark · Sets up equation y = x² + 3 and rearranges to find inverse

A2 (b(ii)) · 1 mark · Expresses inverse as f^{-1}(y) = \sqrt{y-3}, or equivalent (follow-through)

B1 (b(ii)) · 1 mark · States correct domain for inverse (y ≥ 3 or x > 0)

M3 (c) · 1 mark · Uses inverse on 27: \sqrt{24}

A3 (c) · 1 mark · Evaluates \sqrt{24} = 4.90 (2 d.p.) (follow-through)

B2 (c) · 2 marks · Concludes correctly

Worked solution

a) 1 m² = 10,000 cm²; 24000 ÷ 10000 = 2.4 m²

b) i) f(5) = 5² + 3 = 25 + 3 = 28

ii) y = x² + 3 ⇒ x² = y − 3 ⇒ x = sqrt(y − 3); So f^{-1}(y) = sqrt(y − 3), valid for y ≥ 3.

c) f(x) = 27 ⇒ x² + 3 = 27 ⇒ x² = 24 ⇒ x = sqrt(24) ≈ 4.90 (2 d.p.)