SPM 2022 Add Maths Paper 2, Question 1Functions

Solved on video in English and Bahasa Melayu · 2 marks · Form 4, Functions · 4:57 video

The question

(a)

Functions f and g are defined as f: x ↦ 1 − 2x and g: x ↦ x² − 3 respectively.

(a) Determine whether g⁻¹(x) is a function or not. Give a reason for your answer.

[2 marks]

(b)

(b) Find (i) h(x), if f⁻¹h(x) = g(x), (ii) the values of x, if gf⁻¹(x) = 1.

Step 1

g(x) = x² − 3 is a many-to-one function: for example g(2) = 1 and g(−2) = 1, so two objects give the same image.

Why this step

Check g first: if two inputs share one output, reversing it sends that output back to two inputs. That is what breaks the function rule.

4 more steps, each with the reason for it, in the full solution

Final answer

Not a function: g is many-to-one, so g⁻¹(x) = ±√(x + 3) is one-to-many

Where students lose marks

Watch out: 'it has two roots' is not enough. Explain that g is many-to-one, so g⁻¹ is one-to-many, which a function cannot be.

12 more steps, each with the reason for it, in the full solution

Final answer

(i) h(x) = 7 − 2x²; (ii) x = −3 or x = 5

Where students lose marks

Watch out: write f⁻¹ in simplest form, (1 − x)/2, and copy g(x) = x² − 3 carefully when substituting.

Watch every step of this question solved on video (4:57), with the reason behind each step and a box that checks your own answer.

No card. Every past-year question on video, in English and BM.