Functions

Domains

The domain of a function $f$ is the bounds of the values of $x$ for which $f(x)$ is defined. This is most commonly applied when we have a rational function where, for some value of $x$, the denominator will be 0, making $f(x)$ undefined.

Let’s look at the example below:

$f(x) = \dfrac{1}{x^2-x}$

If we rewrite this as the following:

$\dfrac{1}{x(x-1)}$

Then we can more easily see that the denominator will be 0 when $x = 0$ or $x = 1$. Therefore, the domain of $f$ can be written in either of the following notations:

$f \text{ is defined on } \\{x \text{ | } x = \not 0, x = \not 1\\}$

$(-\infty, 0) \cup (0, 1) \cup (1, \infty)$

Continuity

This section requires a relative knowledge of limits, it might help to read through the next topic if any of the notation below is unfamiliar.

A function is said to be continuous at $a$ if:

$\displaystyle\lim_{x \to a} f(x) = f(a)$

This definition can be used to prove a point is continuous if it fulfills the following three properties:

1. $f(a)$ is defined, it is in the domain of $f$

2. $\displaystyle\lim_{x \to a} f(x)$ exists

3. $\displaystyle\lim_{x \to a} f(x) = f(a)$

If a function is not found to be continuous at $a$ then we say that ’$f$ is discontinuous at $a$'.

Odd & even functions

Odd and even functions are mathematical categories of functions, which relate a function to its symmetric properties.

Thinking graphically, even functions can be described as being symmetric in the y-axis.

Even functions can be proven by showing:

$f(-x) = f(x)$

Thinking graphically, odd functions can be described as being symmetric about the origin.

Odd functions can be proven by showing:

$f(-x) = -f(x)$

The only function that is both odd and even is one that equates to 0.

Composite functions

Given two functions $f$ and $g$, the composite function $f \circ g$ is defined by:

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

Be aware of the order in which this is read, the functions are applied from right to left. Just remember the one closest to the $x$ is always applied first.