A function $f$ is called an algebraic function if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) starting with polynomials. Any rational function is automatically an algebraic function. For Example:
$$ f(x)=\sqrt{x^2+1} $$
$$ g(x)=\frac{x^4-16x^2}{x+\sqrt{x}}+(x-2)\sqrt[3]{x+1} $$
A function $P$ is called a polynomial if
$$ P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\dots+a_{2}x^{2}+a_{1}x+a_{0} $$
where $n$ is a nonnegative integer and the numbers $a_0, a_1, a_2, \dots , a_n$ are constants called the coefficients of the polynomial.
Polynomials are fundamental in calculus because their graphs are always smooth (no sharp corners) and continuous (no breaks or holes). Their domain is always the set of all real numbers:
$$ D=(-\infty, +\infty) \text{ or } D=\mathbb{R}. $$
For example, the function
$$ P(x)=2x^6-2x^4+\frac{2}{5}x^3+\sqrt{2} $$
is a polynomial of degree $6$.
There are multiple types of Polynomial functions, most common are:
When we say that $y$ is a linear function of $x$, we mean that the graph of the function is a line, so we can use the slope-intercept form of the equation of a line to write a formula for the function as
$$ y = f(x) = mx + b $$
where $m$ is the slope of the line and $b$ is the $y$-intercept.
Domain of Linear Functions: $D=(-\infty, +\infty)$