SPM 2021 Add Maths Paper 1, Question 13Functions

Solved on video in English and Bahasa Melayu · Form 4, Functions · 5:33 video

The question

(a)

It is given that f(x) = 3/(4x − 1), x ≠ 1/4, and fg(x) = 3/(4x² + 3). Find the object of f(x) if its image is 1/3.

(b)

Express m in terms of p if f(m + 1) = 3 fg(p), where f(x) = 3/(4x − 1) and fg(x) = 3/(4x² + 3).

(c)(i)

Find g(x), given f(x) = 3/(4x − 1) and fg(x) = 3/(4x² + 3).

(c)(ii)

Hence, determine whether the inverse function of g(x) exists or not. Give your justification. (g(x) = x² + 1 from part (c)(i).)

SPM 2021 Add Maths Paper 1, Question 13: the question as drawn in Eduly's video5:33

Part (a)

Step 1

The image is the value of f(x); set f(x) = 1/3.

Why this step

The image is the output of f, so 1/3 replaces f(x), not x. Putting x = 1/3 into f mixes up object and image, the common mistake.

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

Final answer

x = 5/2

Where students lose marks

Watch out: the image is the output. Set f(x) = 1/3 and solve for x; don't substitute x = 1/3 into f.

Part (b)

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

Final answer

m = (1/3)p² − 1/2

Where students lose marks

Watch out: simplify your final answer. Write m = (1/3)p² − 1/2, not an unsimplified fraction like (12p² − 18)/36.

Part (c)(i)

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

Final answer

g(x) = x² + 1

Where students lose marks

Watch out: don't slip on the arithmetic. From 4g(x) − 1 = 4x² + 3, add 1 then divide everything by 4.

Part (c)(ii)

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

Final answer

No: g is many-to-one (g(2) = g(−2) = 5), so g⁻¹ does not exist

Where students lose marks

Watch out: just saying no inverse isn't enough. Show two objects with the same image, e.g. g(2) = g(−2) = 5, so g is many-to-one.

Watch every step of this question solved on video (5:33), with the reason behind each step and a box that checks your own answer.

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