IMAT Mathematics · Functions

Functions, Domain, Range and Composition

A function gives exactly one output for each allowed input; the allowed inputs are the domain, the outputs are the range.

All Functions notes

Why this matters

The 2024 paper asked for the value of a logarithmic function at a point and then for its reciprocal. Domain conditions also decide the right option in inverse-function items, as in 2023.

Concept 1 of 2: Function, domain and range

Think of a function as a machine: put in a number, get out exactly one number. Some inputs break the machine, such as dividing by zero or taking the logarithm of a negative number. The domain is the set of inputs that work; the range is the set of outputs the machine can actually produce.

Definition

A function $f$ assigns to each input $x$ exactly one output $f(x)$ .

The domain is the set of allowed inputs. Remove any $x$ that makes a denominator zero, puts a negative number under a square root, or makes the argument of a logarithm zero or negative.
The range is the set of outputs. For example $x^2 + 4$ is never below 4, so its range is $y \ge 4$ .
To evaluate $f(a)$ , replace every $x$ by $a$ , with brackets.

Worked example

Let

f(x) = \sqrt{x - 3} + \dfrac{1}{x - 5}

. Find the domain of

f

and the value of

f(7)

Practice this conceptself-check · 4 quick reps

The argument of a logarithm must be strictly positive

\ln 0

does not exist, so the condition is

> 0

, not

\ge 0

. A square root is different:

\sqrt{0} = 0

is allowed, so its condition is

\ge 0

The reciprocal of f(a) is a number, found after evaluating

The reciprocal of

f(a)

1/f(a)

: first work out

f(a)

, then turn it upside down. It is not

f(1/a)

, and it is not the inverse function.

Concept 2 of 2: Composite functions: one function after another

A composite function feeds the output of one machine straight into another. The order matters, because adding then squaring is not the same as squaring then adding. In

f(g(x))

the function nearest to

x

, here

g

, acts first.

Definition

The composite $f \circ g$ is defined by $(f \circ g)(x) = f(g(x))$ : apply $g$ first, then $f$ .

In general $f(g(x)) \ne g(f(x))$ .
To find $f(g(x))$ as a formula, write $f$ and replace each $x$ in it by the whole expression $g(x)$ , in brackets.
$f(g(x))$ exists only where $g(x)$ lies in the domain of $f$ .

Composite function

(f \circ g)(x) = f\big(g(x)\big)

$g$ the inner function, applied first
$f$ the outer function, applied to the result

Worked example

Let

f(x) = 2x + 1

and

g(x) = x^2

. Find

f(g(3))

g(f(3))

, and formulas for

f(g(x))

and

g(f(x))

Practice this conceptself-check · 4 quick reps

In f(g(x)) the inner function g acts first

Read

f(g(x))

from the inside out:

g

first, then

f

. Working left to right gives

g(f(x))

, which is usually a different function, and IMAT puts both among the options.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

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

Drill the past-year questions on this topic

Open the bank with this topic pre-filtered.

Open in the bank