Functions

A function is a rule that assigns to each input value exactly one output value. We often write $y = f(x)$, where $x$ is the input and $y$ is the output. The set of all permissible input values is called the domain. The set of all resulting output values is called the range.

When determining the domain, we must be careful about two common restrictions: we cannot divide by zero, and we cannot take the square root of a negative number in the real number system. For example, for $f(x) = \frac{1}{x-2}$, the domain is all real numbers except $x=2$, written as $x \in \mathbb{R}, x \ne 2$. For $g(x) = \sqrt{x}$, the domain is $x \ge 0$.

A composite function is created when one function is applied to the result of another. The notation $fg(x)$ means we first apply the function $g$ to $x$, and then apply the function $f$ to the output, $g(x)$. It's important to work from the inside out. Note that in general, $fg(x)$ is not the same as $gf(x)$.

fg(x) = f(g(x))

An inverse function, denoted $f^{-1}(x)$, 'reverses' the action of the original function $f(x)$. If $f(a) = b$, then $f^{-1}(b) = a$. A function can only have an inverse if it is one-to-one, meaning every output corresponds to a unique input. We can test for this graphically using the horizontal line test: if any horizontal line cuts the graph more than once, the function is not one-to-one and does not have an inverse over that domain.