# BASIC FUNCTIONS

** Linear Functions **

In mathematics, the term **linear function** refers to two distinct but related notions:

- In calculus and related areas, a linear function is a function whose graph is a straight line, that is a polynomial function of degree at most one.
- In linear algebra and functional analysis, a linear function is a
^{ }linear map.

Linear functions are those whose graph is a straight line.

A linear function has the following form

** y = f(x) = a + bx **

A linear function has one independent variable and one dependent variable. The independent variable is **x** and the dependent variable is **y**.

**a** is the constant term or the y intercept. It is the value of the dependent variable when **x = 0**.

**b** is the coefficient of the independent variable. It is also known as the slope and gives the rate of change of the dependent variable.

** Q****uadratic Functions **

A **quadratic function** is one of the form f(x) = ax^{2} + bx + c, where a, b, and c are numbers with a not equal to zero. The graph of a **quadratic function** is a curve called a parabola. ... All parabolas are symmetric with respect to a line called the axis of symmetry.

In algebra, a **quadratic function**, a **quadratic polynomial**, a **polynomial of degree 2**, or simply a **quadratic**, is a polynomial function in one or more variables in which the highest-degree term is of the second degree. For example, a quadratic function in three variables *x*, *y,* and *z* contains exclusively terms *x*^{2}, *y*^{2}, *z*^{2}, *xy*, *xz*, *yz*, *x*, *y*, *z*, and a constant:

with at least one of the coefficients *a, b, c, d, e,* or *f* of the second-degree terms being non-zero.

** Power Functions **

A power function is a function of the form,

* f*(*x*) = **ax**^{p }

where *a* ≠ 0 is a constant and *p* is a real number. Some examples of power functions include:

**Polynomial Functions **

In mathematics, a **polynomial** is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate *x* is *x*^{2} − 4*x* + 7. An example in three variables is *x*^{3} + 2*xyz*^{2} − *yz* + 1.

A ** polynomial function** is a function that can be defined by evaluating a polynomial. More precisely, a function

*f*of one argument from a given domain is a polynomial function if there exists a polynomial that evaluates to

**for all**

*f(x)**x*in the domain of

*f*(here,

*n*is a non-negative integer and

*a*

_{0},

*a*

_{1},

*a*

_{2}, ...,

*a*are constant coefficients). A

_{n}*is a*

**polynomial function****function**such as a quadratic, a cubic, a quartic, and so on, involving only non-negative integer powers of x. We can give a general defintion of a

**polynomial**, and define its degree.

** Rational Functions **

A function ** f(x)** is called a rational function if and only if it can be written in the form

where ** P** and

**are polynomials in**

*Q***and**

*x***is not the zero polynomial. The domain of**

*Q*

**is the set of all points**

*f***for which the denominator**

*x***is not zero.**

*Q(x)*However, if ** P** and

**have a non-constant polynomial greatest common divisor**

*Q***, then setting**

*R***and**

*P = P*_{1}R**produces a rational function which may have a larger domain than**

*Q = Q*_{1}R**, and is equal to**

*f(x)***on the domain of**

*f(x)***. It is a common usage to identify**

*f(x)***and**

*f(x)*

**, that is to extend "by continuity" the domain of**

*f*_{1}(x)**to that of**

*f(x)**Indeed, one can define a rational fraction as an equivalence class of fractions of polynomials, where two fractions In this case*

**f**._{1}(x)A **proper rational function** is a rational function in which the degree of *P(x)* is no greater than the degree of *Q(x)* and both are real polynomials.

** Exponential Functions **

In mathematics, an **exponential function** is a function of the form ** f(x) = b^{x}** in which the argument

*x*occurs as an exponent. A function of the form

**where**

*f(x) = b*^{x+c}**is a constant, is also considered an exponential function and can be rewritten as**

*c***with**

*f(x) = ab*^{x}**.**

*a = b*^{c}As functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function (that is, its derivative) is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base *b*:

The constant * e = 2.71828...* is the unique base for which the constant of proportionality is 1, so that the function's derivative is itself:

** Logarithmic Functions **

In mathematics, the **logarithm** is the inverse function to Exponentiation. That means the logarithm of a given number *x* is the exponent to which another fixed number, the *base* *b*, must be raised, to produce that number *x*. In the simplest case the logarithm counts repeated multiplication of the same factor; e.g., since 1000 = 10 × 10 × 10 = 10^{3}, the "logarithm to base 10" of 1000 is 3. The logarithm of *x* to *base* *b* is denoted as log_{b} (*x*) (or, without parentheses, as log_{b} *x*, or even without explicit base as log *x*, when no confusion is possible). More generally, exponentiation allows any positive real number to be raised to any real power, always producing a positive result, so the logarithm for any two positive real numbers *b* and *x* where *b* is not equal to 1, is always a unique real number *y*. More explicitly, the defining relation between exponentiation and logarithm is:

exactly if

**For example**, **log _{2} 32 = 5, as 32 = 2^{5}**.

The logarithm to base 10 (that is *b* = 10) is called the common logarithm and has many applications in science and engineering. The natural logarithm has the number * e* (that is

*b*≈ 2.718) as its base; its use is widespread in mathematics and physics, because of its simpler derivative. The binary logarithm uses base 2 (that is

*b*= 2) and is commonly used in computer science.

Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations. They were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. Tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition because of the fact—important in its own right—that the logarithm of a product is the sum of the logarithms of the factors:

provided that *b*, *x* and *y* are all positive and *b* ≠ 1. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century.

** Sinusoidal Functions **

A **sinusoidal function **(or **sinusoidal oscillation **or **sinusoidal signal**) is one that can be wrtten in the for*m *** f (t) = A cos(ωt − φ)**.

The function

*f*(

*t*) is a cosine function which has been

**amplified**by

*A*,

**shifted**by

*φ*/

*ω*, and

**compressed**by

*ω*.

- A>0 is its
**amplitude**: how high the graph of*f*(*t*) rises above the*t*-axis at its maximum values; *φ*is its**phase lag**: the value of*ωt*for which the graph has its maximum (if*φ*= 0, the graph has the position of cos(*ωt*); if*φ*=*π*/2, it has the position of sin(*ωt*));-
*T*=*φ*/*ω*is its**time delay**or**time lag**: how far along the*t*-axis the graph of cos(*ωt*) has been shifted to make the graph of the above function; (to see this, write*A*cos(*ωt*−*φ*) =*A*cos(*ω*(*t*−*φ*/*ω*))) *ω*is its**angular frequency**: the number of complete oscillations*f*(*t*) makes in a time interval of length 2*π*; that is, the number of radians per unit time;*v*=*ω/*2*π*is the**frequency**of*f*(*t*): the number of complete oscillations the graph makes in a time interval of length 1; that is, the number of cycles per unit time;- P=2
*π/**ω=1/**v*is its**period**, the*t*-interval required for one complete oscillation.

One can also write the function using the time lag *τ* = *φ*/*ω*

** f (t) = A cos (_{ω}(t − _{τ})) **.

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