Functions

Functions is the opening topic of the 0606 syllabus and a reliable 6-10 marks in most sessions. The mathematics is friendly; the marks are lost to notation and to range/domain reasoning done by guesswork instead of method.

Domain and range: reason from a sketch, not a hunch

The domain is the set of allowed inputs; the range is the set of outputs actually produced. For every range question, the method is the same: sketch or visualise the graph over the given domain, then read the outputs. For quadratics, complete the square first, the vertex gives the extreme value (full method in the quadratics notes). Write ranges in proper notation: $f(x) \ge 3$ or $-1 < f(x) \le 5$, using the function’s letter, not $x$.

Watch for restricted domains: the range of $f(x) = x^2$ for $x \ge 2$ is $f(x) \ge 4$, not $f(x) \ge 0$. Examiners set exactly this trap.

Composite functions: right to left, with brackets

$fg(x)$ means $g$ first, then $f$, substitute the whole of $g(x)$ wherever $x$ appears in $f$:

$f(x) = 2x - 1$, $g(x) = x^2 + 3$ $fg(x) = 2(x^2 + 3) - 1 = 2x^2 + 5$ $gf(x) = (2x - 1)^2 + 3 = 4x^2 - 4x + 4$

Method-mark habits: write the substitution line with brackets intact before expanding. Note $gf \ne fg$ in general, if your two composites match, recheck. For equations like $fg(x) = 7$, build the composite first, then solve.

Inverse functions: the four-step routine

To find $f^{-1}$ for $f(x) = \frac{3x + 2}{x - 1}$:

1. Write $y = \frac{3x + 2}{x - 1}$
2. Swap the roles: make $x$ the subject, multiply up: $y(x - 1) = 3x + 2 \to x(y - 3) = y + 2 \to x = \frac{y + 2}{y - 3}$
3. Exchange letters: $f^{-1}(x) = \frac{x + 2}{x - 3}$
4. State domain if asked: domain of $f^{-1} =$ range of $f$

Every line of that rearrangement is auditable working, the multiply-up and collect-$x$ steps each carry method credit. Graphically, $y = f^{-1}(x)$ is $y = f(x)$ reflected in $y = x$; sketch questions award a B mark for showing that symmetry, so draw the dashed $y = x$ line.

Modulus functions and their graphs

$|f(x)|$ reflects every below-axis part of the graph upward. For $y = |2x - 4|$: sketch $y = 2x - 4$, fold the negative section up, label the vertex $(2, 0)$ and intercept $(0, 4)$, labelled features are where the sketch marks live. Solving modulus equations and inequalities has its own dedicated methods in equations, inequalities and graphs.

One-one functions and why they matter

A function is one-one if each output comes from exactly one input, the horizontal-line test. Only one-one functions have inverses, which is why 0606 questions restrict domains: $f(x) = (x - 3)^2$ is not one-one on $\mathbb{R}$, but on $x \ge 3$ it is, and the inverse exists. The exam phrasing “explain why $f$ has an inverse” wants the words one-one in the answer; “state the smallest value of $k$ such that $f$ is one-one for $x \ge k$” wants the vertex $x$-coordinate.

Worked exam-style question

$f(x) = x^2 - 4x + 7$ for $x \ge k$. (i) Find the smallest $k$ for which $f$ is one-one. (ii) For this $k$, find $f^{-1}(x)$.

(i) Complete the square: $f(x) = (x - 2)^2 + 3$. Vertex at $x = 2$, so smallest $k = 2$. (ii) $y = (x - 2)^2 + 3 \to (x - 2)^2 = y - 3 \to x - 2 = \sqrt{y - 3}$ (positive root since $x \ge 2$, state this) $\to x = 2 + \sqrt{y - 3}$. $f^{-1}(x) = 2 + \sqrt{x - 3}$, for $x \ge 3$.

The mark scheme rewards: completed-square form (M), $k = 2$ (A), the rearrangement (M), choosing the positive root with reason (B), final inverse (A). The root-choice sentence is the most-missed mark.

Common mistakes in this topic

• Computing $fg$ as $gf$ (order reversal)
• Range given in $x$ instead of $f(x)$ notation
• Ignoring the restricted domain when stating a range
• Dropping the $\pm$ reasoning when inverting a square, then losing the final mark
• Treating $f^{-1}(x)$ as $\frac{1}{f(x)}$: the reciprocal is not the inverse

For the full marking logic behind M/A/B credits, see how to show working for full marks; functions also feed directly into quadratics and logs & exponentials, where inverse reasoning returns.

Stuck on inverse domains or modulus sketches? That’s a one-lesson fix in most cases, book a free 1-hour trial class with Teacher Rig on WhatsApp.