IMAT Mathematics · Free notes

Functions: Domain, Graphs and Inverses

Functions has 5 past questions since 2011, and the ministry papers from 2023 on have asked 4 of them. So the topic now comes up about once a year, and each item is short: evaluate a function, find where a quadratic turns or which point it passes through, invert a logarithm or an exponential, or say where a function is positive. The difficulty sits in the details: the domain of a logarithm, the sign inside a bracket, and the difference between an inverse and a reciprocal. Know the shapes of the standard graphs by heart and most of these items take a minute.

The handout has every subtopic, worked example, formula and trap with answers shown. The formula sheet is the chapter’s formulas and traps alone, on one page.

IMAT home All formulas on one page

Subtopic notes

Formula & revision sheet

5 formulas · 1 reference tables · 9 gotchas across all subtopics — the exam-eve cheat-sheet

Functions, Domain, Range and Composition

Formulas (1)

Composite functions: one function after another · Composite function $(f \circ g)(x) = f\big(g(x)\big)$

Watch out for (3)

The argument of a logarithm must be strictly positive
→ Function, domain and range
The reciprocal of f(a) is a number, found after evaluating
→ Function, domain and range
In f(g(x)) the inner function g acts first
→ Composite functions: one function after another

Graphs of Standard Functions

Formulas (2)

Linear and quadratic graphs: gradient, vertex and roots · Vertex of y = ax² + bx + c $x_v = -\frac{b}{2a}, \qquad y_v = a x_v^2 + b x_v + c$
Even and odd functions, increasing and decreasing, reading a graph · Even and odd $\text{even: } f(-x) = f(x) \qquad \text{odd: } f(-x) = -f(x)$

Reference tables (1)

Shapes of power, reciprocal, exponential and logarithmic graphs6 rows

Function	Domain	Range	Shape and key points
$y = x^2$ (even powers)	All real x	$y \ge 0$	U shape, symmetric about the y-axis, through $(0, 0)$ and $(1, 1)$
$y = x^3$ (odd powers)	All real x	All real y	Rises from left to right, symmetric about the origin, through $(0, 0)$ and $(1, 1)$
$y = \sqrt{x}$	$x \ge 0$	$y \ge 0$	Starts at $(0, 0)$ and rises ever more slowly
$y = \dfrac{1}{x}$	$x \ne 0$	$y \ne 0$	Two separate branches; both axes are asymptotes
$y = a^x$ with $a > 1$ , e.g. $e^x$	All real x	$y > 0$	Through $(0, 1)$ ; increasing; the x-axis is an asymptote on the left An exponential is positive for every real x, including negative x.
$y = \log_a x$ with $a > 1$ , e.g. $\ln x$	$x > 0$	All real y	Through $(1, 0)$ ; increasing; the y-axis is an asymptote A logarithm is negative for $0 < x < 1$ and positive for $x > 1$ .

Exponential and logarithm with the same base are inverse functions, so their graphs are reflections of each other in the line

y = x

Watch out for (3)

The axis is at minus b over 2a
→ Linear and quadratic graphs: gradient, vertex and roots
An exponential is never zero or negative
→ Shapes of power, reciprocal, exponential and logarithmic graphs
A constant term keeps a function even but stops it being odd
→ Even and odd functions, increasing and decreasing, reading a graph

Inverse Functions and Transforming Graphs

Formulas (2)

Inverse functions: undoing a function · Inverse function $y = f(x) \quad\Rightarrow\quad x = f^{-1}(y)$
Transformations of graphs: shifts, stretches and reflections · Stretch, then shift $y = a\, f(x - h) + k$

Watch out for (3)

The inverse function is not the reciprocal
→ Inverse functions: undoing a function
Give the inverse with its domain
→ Inverse functions: undoing a function
f(x minus 3) moves the graph right, not left
→ Transformations of graphs: shifts, stretches and reflections

PYQ weightage by concept

7 concepts · 0 PYQs — where the marks actually sit, so you know what to drill first

Functions, Domain, Range and Composition0 PYQs · 0%

Concept	PYQs	Share
Function, domain and rangefoundation	—	—
Composite functions: one function after anotherfoundation	—	—

Graphs of Standard Functions0 PYQs · 0%

Concept	PYQs	Share
Linear and quadratic graphs: gradient, vertex and rootsfoundation	—	—
Shapes of power, reciprocal, exponential and logarithmic graphsfoundation	—	—
Even and odd functions, increasing and decreasing, reading a graphfoundation	—	—

Inverse Functions and Transforming Graphs0 PYQs · 0%

Concept	PYQs	Share
Inverse functions: undoing a functionfoundation	—	—
Transformations of graphs: shifts, stretches and reflectionsfoundation	—	—