## 1-2 : Mathematical Models: A Catalog of Essential Functions.

A mathematical model is mathematical description of real-world phenomenon, the model is to understand the phenomenon and perhaps to make predictions about future behavior.

There are many different types of functions that can be used to model relationships observed in the real word.

a- Linear Models $\mapsto y=f(x)=mx+b$
where: $m$  slope of the line, $b$  is the y– intercept.

b- Polynomials$\mapsto P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+...+a_{2}x^{2}+a_{1}x+a_{0}$

where:$a_{1},a_{2},a_{3}...a_{n}$  are constants called the coefficients of the polynomials. The domain of any polynomials is $\mathbb{R}=(-\infty,\infty )$, If  $a_{n}\neq 0$, then the degree of the polynomials is  $n$

c- Power Functions $\mapsto x^{a}$

where : $a$  is a constant

d- Rational Function $\mapsto f(x)=\frac{P(x)}{Q(x)}$

where :$P$ and $Q$  are polynomials, the domain consists of all values of x such that $Q(x)\neq 0$

e- Algebraic  Funcions: If it can be constructed using algebraic operations( such as additions, subtraction, multiplication, division, and taking roots) starting with polynomials. Any rational function is automatically an algebraic function. More example  $f(x)=\sqrt{x^{2}+1}$

f- Trigonometric Functions:$sin(x),cos(x),tan(x)...$

g- Exponential Functions $\mapsto f(x)=a^{x}$

where :$a$  is a positive constant.

h- Logarithmic Functions$\mapsto log_{a}x$

where:  the base $a$  is a positive constant.

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