Functions — IMAT Mathematics Formulas and Traps

Functions, Domain, Range and Composition

Composite functions: one function after another

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

Common traps

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)

is

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.

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.

Graphs of Standard Functions

Learn this subtopic in the notes

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

$x_v$ x-coordinate of the vertex (the axis of symmetry)
$y_v$ y-coordinate of the vertex: substitute $x_v$

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)

Shapes of power, reciprocal, exponential and logarithmic graphs

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

.

Common traps

The axis is at minus b over 2a

The vertex is at

x = -b/(2a)

. Dropping the minus sign puts the vertex on the wrong side of the y-axis, and forgetting the 2 doubles its distance from the axis. Both wrong points appear as options.

An exponential is never zero or negative

e^x > 0

for every real

x

: at

x = -10

it is tiny but still positive. Options such as "only for

x > 0

" or "only for

x \ge 1

" confuse

e^x > 0

with

e^x > 1

, which holds only for

x > 0

.

A constant term keeps a function even but stops it being odd

Adding a constant to an even function leaves it even (

x^2 + 7

is even). Adding a nonzero constant to an odd function makes it neither (

x^3 + 1

), because an odd function defined at 0 must pass through the origin.

Inverse Functions and Transforming Graphs

Learn this subtopic in the notes

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

$a$ vertical stretch factor (a reflection in the x-axis if negative)
$h$ shift to the right
$k$ shift upwards

Common traps

The inverse function is not the reciprocal

f^{-1}(x)

means the function that undoes

f

;

\dfrac{1}{f(x)}

is the reciprocal, a different thing. For

f(x) = 2x

,

f^{-1}(x) = x/2

but

1/f(x) = 1/(2x)

.

Give the inverse with its domain

An option can have the right formula and the wrong domain. The domain of

f^{-1}

is the range of

f

: work out which outputs

f

really produces before choosing.

f(x minus 3) moves the graph right, not left

A change inside the bracket works against its sign:

f(x - 3)

is the graph moved 3 units right,

f(x + 3)

is it moved 3 units left. Outside the bracket the sign means what it says:

f(x) - 3

is 3 units down.

Functions formulas

Functions, Domain, Range and Composition

Composite functions: one function after another

Common traps

Graphs of Standard Functions

Linear and quadratic graphs: gradient, vertex and roots

Even and odd functions, increasing and decreasing, reading a graph

Shapes of power, reciprocal, exponential and logarithmic graphs

Common traps

Inverse Functions and Transforming Graphs

Inverse functions: undoing a function

Transformations of graphs: shifts, stretches and reflections

Common traps

More IMAT Mathematics formula sheets