Form 4 Chapter 1: Functions (Fungsi)

SPM KSSM Diagnostic Drill

Total Marks: 40 Marks • Time Allowed: 50 Minutes

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A Bahagian A: SPM Paper 1 Format [16 Marks]

Answer all questions

Soalan 1 [3 Marks]

Diagram 1 shows the relation between set $X$ and set $Y$.

(a) State the type of relation. [1 mark]

(b) Using function notation, represent this relation. [1 mark]

(c) State whether the inverse of this relation is a function. Give your reason. [1 mark]

Soalan 2 [5 Marks]

Given the function $f(x) = \frac{3x + 1}{x - 2}, x \neq 2$ and $g(x) = kx - 3$, where $k$ is a constant.

(a) Find $f^{-1}(x)$ and state the value of $x$ for which $f^{-1}(x)$ does not exist. [3 marks]

(b) Given that $gf(3) = 17$, find the value of $k$. [2 marks]

Soalan 3 (Paper 1 Sec B) [8 Marks]

A function $f$ is defined by $f: x \mapsto |2x - 6|$ for domain $-1 \le x \le 5$.

(a) Sketch the graph of $f(x)$ on the given domain and state the range of $f(x)$. [3 marks]

(b) Find the values of $x$ such that $f(x) = 4$. [2 marks]

(c) Determine the values of constant $m$ such that the equation $f(x) = mx$ has exactly two distinct solutions. [3 marks]

B Bahagian B: SPM Paper 2 Format & Real-World KBAT [24 Marks]

Answer all questions

Soalan 4 (Paper 2 Standard) [10 Marks]

Given the function $f(x) = 5 - 2x$ and the composite function $gf(x) = \frac{10 - 4x}{3 - 2x}, x \neq \frac{3}{2}$.

(a) Find the function $g(x)$ in terms of $x$. [3 marks]

(b) Find $g^{-1}(x)$ and state the domain where $g^{-1}(x)$ is defined. [3 marks]

(c) Find the function $h(x)$ such that $hg(x) = 4x + 1$. [2 marks]

(d) Calculate the value of $x$ such that $f^2(x) = g(1)$. [2 marks]

Soalan 5 (KBAT / HOTS Application) [8 Marks]

A pharmaceutical company develops a temperature-sensitive vaccine storage system. The inner chamber temperature $T$ (in °C) after $t$ hours of operation is modeled by the function:

$T(t) = 30 - 4t, \quad 0 \le t \le 8$

The biological potency index $P$ of the vaccine depends on chamber temperature $T$ through the function:

$P(T) = \frac{100}{T^2 - 4T + 20}$

(a) State the meaning of the inverse function $t = T^{-1}(T)$ in this operational context, and derive an expression for $T^{-1}(T)$. [2 marks]

(b) Express the potency index $P$ directly as a composite function of operating hours $t$, $P(t) = PT(t)$. [2 marks]

(c) Determine the operating hour $t$ at which the vaccine achieves its maximum potency index. Calculate this maximum potency value. [4 marks]

Soalan 6 (Non-Routine Synthesis) [6 Marks]

A function $f$ satisfies $f(x) = \frac{x + 1}{x - 1}, x \neq 1$.

(a) Show that $f^2(x) = x$ and deduce the value of $f^{2026}(5)$. [3 marks]

(b) Hence, evaluate the product $P = f(2) \cdot f(3) \cdot f(4) \cdots f(100)$. [3 marks]