Definition 1.2.1. A function $f:\mathbb R \to \mathbb R$ is a polynomial function if there exist numbers $n\in \mathbb N$ and $c_n,\dots, c_0\in \mathbb R$ such that for every input $x\in \mathbb R$
\[f(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0.\]Definition 1.2.2. A function $f:D\to\mathbb{R}$ is a rational function if there exist polynomial functions $p,q:\mathbb{R}\to\mathbb{R}$, with $q$ not the zero polynomial function such that
\[f(x)=\frac{p(x)}{q(x)}\]for every input $x\in D$, where $D=\lbrace x\in\mathbb{R}:q(x)\neq 0\rbrace.$
Definition 1.2.3. A function $f:D\to\mathbb{R}$ is a root function if there exists a positive integer $n$ such that
\[f(x)=\sqrt[n]{x}\]for every input $x\in D$, where $D=[0,\infty)$ if $n$ is even and $D=\mathbb{R}$ if $n$ is odd.
Definition 1.2.4. A function $f$ is an algebraic function if it can be constructed using algebraic operations—such as addition, subtraction, multiplication, division, and taking roots—starting with polynomials. And, a function $f$ is a transcendental function if it can’t be constructed using the algebraic operations.
Definition 1.2.5 (Trigonometric Functions). For each $x\in \mathbb R$, let $P_x=(a(x),b(x))$ be the point on the unit circle obtained by moving a signed arc length $x$ from the point $(1,0)$, counterclockwise if $x>0$ and clockwise if $x<0$. The sine function and the cosine function are the functions $\sin,\cos:\mathbb R\to\mathbb R$ such that
\[\sin x=b(x),\]and
\[\cos x=a(x)\]for every $x\in\mathbb R$. The tangent function is the function $\tan:D\to\mathbb R$ such that
\[\tan x=\frac{\sin x}{\cos x}\]for every $x\in D$, where $D=\lbrace x\in\mathbb R:\cos x\neq 0\rbrace .$