SPM High-Frequency Core Topic

Mastering Functions & Inverses

Functions form the bedrock of SPM Additional Mathematics. This chapter explores relations, one-to-one tests, composite functions $f(g(x))$, inverse functions $f^{-1}(x)$, self-inverses, and real-world temperature & currency conversion mappings.

1. Core Mathematical Theorems & Formulas

Composite Functions

Operation where the output of function $g$ becomes the input of function $f$:

$fg(x) = f[g(x)]$
  • • In general, $fg(x) \neq gf(x)$ (not commutative).
  • • Composite of 3 functions: $fgh(x) = f(g(h(x)))$.
  • • Self-composition: $f^2(x) = f(f(x))$, $f^3(x) = f(f(f(x)))$.

Inverse Functions

Reverses the effect of function $f$. Exists if and only if $f$ is a one-to-one function (passes Horizontal Line Test):

$ff^{-1}(x) = f^{-1}f(x) = x$
  • • Domain of $f^{-1} = \text{Range of } f$.
  • • Range of $f^{-1} = \text{Domain of } f$.
  • • Graph of $y = f^{-1}(x)$ is a reflection of $y = f(x)$ on line $y = x$.
Absolute Value Function: $f(x) = |ax + b| = \begin{cases} ax + b, & \text{if } ax + b \ge 0 \\ -(ax + b), & \text{if } ax + b < 0 \end{cases}$. The graph is always $\ge 0$ (V-shaped or reflected). It is many-to-one over its whole domain, hence $f^{-1}$ does NOT exist unless domain is restricted!

2. SPM Examiner Pitfalls & Frequent Blunders

Trap 1: Confusing Inverse Notation with Exponents:

$f^{-1}(x) \neq \frac{1}{f(x)}$ and $[f(x)]^{-1} = \frac{1}{f(x)}$. Never treat $-1$ as an algebraic reciprocal power in functions!

Trap 2: Forgetting Domain Restriction in Rational Functions:

When finding $f^{-1}(x)$ for $f(x) = \frac{ax+b}{cx+d}$, you must explicitly state the excluded value: $x \neq -\frac{d}{c}$ for $f$, and $x \neq \frac{a}{c}$ for $f^{-1}$. SPM mark schemes penalize omitting $x \neq k$.

Trap 3: Finding Outer Function vs Inner Function:

If given $g(x)$ and $fg(x)$, finding outer function $f(x)$ requires letting $u = g(x) \implies x = g^{-1}(u)$. If finding inner function $g(x)$ when $f(x)$ and $fg(x)$ are given, apply $f[\dots]$ directly or $g(x) = f^{-1}(fg(x))$. Mixing these up loses 3 method marks ($K1, K1$).

3. Progressive Worked Examples (SPM Standard to KBAT)

Example 1 • Foundation Level (P1 Standard) [4 Marks]
Given the function $f(x) = 3x - 5$ and $g(x) = \frac{2}{x - 1}, x \neq 1$.

(a) Find $gf(x)$ and state the value of $x$ for which $gf(x)$ is undefined. [2 marks]

(b) Find $f^{-1}(4)$. [2 marks]

Step-by-Step Marking Scheme:
(a) $gf(x) = g(3x - 5) = \frac{2}{(3x - 5) - 1} = \frac{2}{3x - 6}$ [K1: Substitute $f(x)$ into $g$]
$gf(x) = \frac{2}{3(x - 2)}$ is undefined when $3x - 6 = 0 \implies x = 2$ [N1: Correct $x=2$]
(b) Let $f(x) = 4 \implies 3x - 5 = 4 \implies 3x = 9$ [K1: Equate $f(x)=4$ or derive $f^{-1}(x)=\frac{x+5}{3}$]
$x = 3 \implies f^{-1}(4) = 3$ [N1: Correct value]
Example 2 • SPM Intermediate (Finding Unknown Function) [4 Marks]
Given $g(x) = 2x + 3$ and $fg(x) = 4x^2 + 12x + 7$. Find:

(a) The function $f(x)$. [3 marks]

(b) The values of $k$ if $f(k) = 23$. [1 mark]

Step-by-Step Marking Scheme:
(a) Let $u = g(x) = 2x + 3 \implies x = \frac{u - 3}{2}$ [K1: Express $x$ in terms of $u$]
$f(u) = 4\left(\frac{u - 3}{2}\right)^2 + 12\left(\frac{u - 3}{2}\right) + 7 = 4\left(\frac{u^2 - 6u + 9}{4}\right) + 6(u - 3) + 7$ [K1: Substitute $x$ into $fg(x)$]
$f(u) = (u^2 - 6u + 9) + (6u - 18) + 7 = u^2 - 2 \implies f(x) = x^2 - 2$ [N1: Correct $f(x) = x^2 - 2$]
(b) $f(k) = k^2 - 2 = 23 \implies k^2 = 25 \implies k = \pm 5$ [N1: Both $\pm 5$ must be stated]
Example 3 • Paper 2 Section A Multi-Part [7 Marks]
The function $h$ is defined by $h(x) = \frac{4x - 1}{x + 2}, x \neq -2$.

(a) Find an expression for $h^{-1}(x)$, stating its domain restriction. [3 marks]

(b) Another function $p$ satisfies $ph(x) = 8x - 2$. Find $p(x)$. [2 marks]

(c) Determine whether the function $h$ has a self-inverse (i.e. $h(x) = h^{-1}(x)$). Justify your answer. [2 marks]

Step-by-Step Marking Scheme:
(a) Let $y = \frac{4x - 1}{x + 2} \implies y(x + 2) = 4x - 1 \implies xy + 2y = 4x - 1$ [K1: Clear fraction & group $x$]
$x(4 - y) = 2y + 1 \implies x = \frac{2y + 1}{4 - y} \implies h^{-1}(x) = \frac{2x + 1}{4 - x}$ [N1: Correct $h^{-1}(x)$]
Domain restriction: $x \neq 4$ [P1: Stated $x \neq 4$]
(b) Notice $ph(x) = 8x - 2 = 2(4x - 1)$. Since $h(x) = \frac{4x-1}{x+2} \implies 4x - 1 = (x+2)h(x)$. [K1: Use relation or $p(u) = 8(h^{-1}(u)) - 2$]
$p(u) = 8\left(\frac{2u+1}{4-u}\right) - 2 = \frac{16u+8 - 8 + 2u}{4-u} = \frac{18u}{4-u} \implies p(x) = \frac{18x}{4-x}, x \neq 4$ [N1: Correct $p(x)$]
(c) If $h(x) = h^{-1}(x)$, then $\frac{4x-1}{x+2} = \frac{2x+1}{4-x}$. Testing coefficients: in general $\frac{ax+b}{cx+d}$ is self-inverse if $a = -d$. Here $a = 4$ and $d = 2$, so $a \neq -d$ ($4 \neq -2$). [K1: Compare $h(x)$ and $h^{-1}(x)$ or check $a = -d$]
Since $\frac{4x-1}{x+2} \neq \frac{2x+1}{4-x}$ identically, $h$ does NOT have a self-inverse. [N1: Valid conclusion]
Example 4 • KBAT Real-World Context (Currency Conversion) [5 Marks]
An e-commerce shopper in Malaysia purchases an electronic gadget from an overseas store. The store charges a price in USD ($x$) plus a shipping fee of $\$15$, defined by the function $p(x) = x + 15$. The Malaysian credit card provider converts USD ($u$) to Malaysian Ringgit (RM) via the function $c(u) = 4.40u + 12$, where $4.40$ is the exchange rate and RM 12 is the bank's international transaction fee.

(a) Formulate a composite function $T(x) = cp(x)$ that computes the total payment in RM for a gadget with USD price $x$. [2 marks]

(b) If the shopper pays a total of RM 540 for the order, determine the retail price $x$ of the gadget in USD. [3 marks]

Step-by-Step Marking Scheme:
(a) $T(x) = c[p(x)] = 4.40(x + 15) + 12$ [K1: Substitute $p(x)$ into $c(u)$]
$T(x) = 4.40x + 66 + 12 \implies T(x) = 4.40x + 78$ [N1: Simplified linear function in RM]
(b) Given $T(x) = 540 \implies 4.40x + 78 = 540$ [K1: Equate composite function to 540]
$4.40x = 540 - 78 = 462 \implies x = \frac{462}{4.40}$ [K1: Algebraic rearrangement]
$x = 105$. The retail price is $\$105\text{ USD}$. [N1: Correct final value with unit]

Scientific Calculator Check: Casio fx-570EX / 991CW

You can rapidly verify inverse function and composite values in Table Mode (MENU → 9 (Table)):

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Open Form 4 Chapter 1 Worksheet