Types of functions. Let $n$ be an integer, and let $a, a_0, a_1, \ldots, a_n, b, m$ be real numbers.

A linear function is of the form $mx+b$ ($m$ is the slope, $b$ is the $y$-intercept)

A polynomial function is of the form $a_n x^n+a_{n-1}x^{n-1}+\cdots+a_2x^2+a_1x+a_0$

A power function is of the form $bx^a$ ($a$ is the fixed exponent or power)

A root function is of the form $x^{1/n}=^n\sqrt{x}$

A rational function is of the form $\frac{p(x)}{q(x)}$, where $p$ and $q$ are polynomials

An algebraic function is a function that can be constructed from polynomials using algebraic operations ($+$, $-$, $\cdot$, $/$, $x^a$) repeatedly.

A basic trigonometric function is one of $\sin x$, $\cos x$, $\tan x$, $\mbox{csc}x$, $\mbox{sec}x$, $\mbox{cot}x$

An exponential function is of the form $a^x$, where $a>0$ (can you see why the base $a$ should be positive?)

A logarithmic function is of the form $\log_a x$, where $a>0$ is the fixed base