Answer
See below.
Work Step by Step
$\text{Linear functions}$
Linear functions are of the form $f(x)=ax+b$,
where a and b are real numbers.
Examples:
$f(x)=-2x+1$
$f(x)=3x$
$\text{Power functions}$
Power functions are of the form $f(x)=x^{a}$,
where a is a constant..
Examples:
$f(x)=x^{3}$
$f(x)=x^{-2}$
$f(x)=x^{1.5}$
$\text{Polynomial functions}$
Polynomial functions are of the form $f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}$,
where $a_{i}$ are real constants and n is a nonnegative integer.
Examples:
$f(x)=3x^{2}+2x-1$
$f(x)=x^{7}-1$
$\text{Rational functions}$
have the form $f(x)=\displaystyle \frac{p(x)}{q(x)}$,
where p and q are polynomials.
Examples:
$f(x)=\displaystyle \frac{x^{2}-1}{x+3},\qquad f(x)=\frac{x^{3}-2x^{2}+x-4}{x^{5}+x-2}$
$\text{Algebraic functions}$
These functions are constructed from polynomials using algebraic operations - adding, subtracting, multiplying, dividing, taking roots.
Examples:
$f(x)=\displaystyle \frac{x-1}{x^{2}}+\sqrt{x-1}$
$f(x)=(x^{2}+1)\sqrt{1-x^{2}}$
$\text{Transcendental functions}$
The functions are functions that are not algebraic.
The next few categories are examples of transcendental functions.
$\text{Trigonometric functions}$
Functions of the sine, cosine, tangent, cotangent, secant, cosesecant.
Examples:
$f(x)=\sin(x)$
$f(x)=\cot(x)$
$\text{Exponential functions}$
are functions of the form $f(x)=b^{x}$,
where the base b is a positive real number.
Examples:
$f(x)=2^{x}$
$f(x)=e^{x}$
$\text{Logarithmic functions}$
These are functions that are inverses of exponential functions.
Examples:
$f(x)=\log_{3}x\qquad $(base is 3)
$f(x)=\ln(x)\qquad $(base is e)
$f(x)=\log x\qquad $(base is 10)
