SPM 2024 Add Maths Paper 1, Question 14Functions

Solved on video in English and Bahasa Melayu · Form 4, Functions · 6:37 video

The question

(a)

Diagram 11.1 shows the relation of three sets. Set A contains x, Set B contains x - 2, and Set C contains x² + 1. The function f maps from Set A to Set B (f: x → x - 2) and the function g maps from Set A to Set C. (a) State gf(x).

(b)(i)

Using the same functions f and gf from part (a) (where f(x) = x - 2 and gf(x) = x² + 1), (b)(i) determine f^{-1}(x). Hence, find g(x).

(b)(ii)

Diagram 11.2 shows part of the graph y = g(x), where g(x) = x² + 4x + 5 (from part b(i)), with y-intercept at (0, p). (b)(ii) Sketch the graph g^{-1}(x) on Diagram 11.2.

(c)

Using f(x) = x - 2 from the same question, (c) find f²(x), f³(x) and f⁴(x). Hence, state f^{75}(x).

Part (a)

Step 1

From Diagram 11.1, the combined mapping gf sends x directly to x² + 1 (read off from the diagram arrow from Set A to Set C).

Why this step

The arrow from Set A to Set C already is gf (f first, then g), so read it off. Swapping f and g is the common mistake.

1 more step, with the reason for it, in the full solution

Final answer

gf(x) = x² + 1

Where students lose marks

Watch out: read the diagram carefully; f is x − 2, so don't swap f and g, which gives the wrong x² − 1.

Part (b)(i)

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

Final answer

f⁻¹(x) = x + 2; g(x) = x² + 4x + 5

Where students lose marks

Watch out: substitute f⁻¹(x) = x + 2, not f(x), into gf. Using f gives the wrong sign: x² − 4x + 5.

Part (b)(ii)

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

Final answer

Sketch (see video), through (5, 0)

Where students lose marks

Watch out: draw the dashed line y = x and label (p, 0) on the x-axis; weaker candidates missed one or both.

Part (c)

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

Final answer

f²(x) = x − 4, f³(x) = x − 6, f⁴(x) = x − 8, f⁷⁵(x) = x − 150

Where students lose marks

Watch out: fⁿ means apply f n times, not a power. Spot the pattern x − 2n instead of squaring.

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