BASIC FUNCTIONS

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.

    Quadratic Functions    

A quadratic function is one of the form f(x) = ax2 + 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 x2, y2, z2, xy, xz, yz, x, y, z, and a constant:

f(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz+gx+hy+iz+j,

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  

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 x2 − 4x + 7. An example in three variables is x3 + 2xyz2yz + 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 f(x) for all x in the domain of f (here, n is a non-negative integer and a0, a1, a2, ..., an are constant coefficients). A polynomial function is a 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 f(x)={\frac {P(x)}{Q(x)}}

where  P and Q  are polynomials in x and Q is not the zero polynomial. The domain of  f is the set of all points  x for which the denominator Q(x) is not zero.

However, if P and Q have a non-constant polynomial greatest common divisor R, then setting P = P1R and Q = Q1R produces a rational function f_{1}(x)={\frac {P_{1}(x)}{Q_{1}(x)}},which may have a larger domain than f(x), and is equal to f(x) on the domain of f(x). It is a common usage to identify f(x) and f1(x), that is to extend "by continuity" the domain of f(x) to that of f1(x). Indeed, one can define a rational fraction as an equivalence class of fractions of polynomials, where two fractions   In this case  

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) = bx in which the argument x occurs as an exponent. A function of the form f(x) = bx+c where c is a constant, is also considered an exponential function and can be rewritten as f(x) = abx with a = bc.

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:

  {\displaystyle {\frac {d}{dx}}b^{x}=b^{x}\log _{e}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:

  {\displaystyle {\frac {d}{dx}}e^{x}=e^{x}\log _{e}e=e^{x}.}
 

   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 = 103, the "logarithm to base 10" of 1000 is 3. The logarithm of x to base b is denoted as logb (x) (or, without parentheses, as logbx, 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:

  {\displaystyle \log _{b}(x)=y\quad } exactly if  {\displaystyle \quad b^{y}=x.}

For example, log2 32 = 5, as 32 = 25.

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:

{\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y,\,}

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 form f (t) = A cos(ωtφ).
The function f (t) is a cosine function which has been amplified by A, shifted by φ/ω, and compressed by ω.

  1. A>0 is its amplitude: how high the graph of f (t) rises above the t-axis at its maximum values;
  2. 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));
  3.  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(ωtφ) = A cos(ω(tφ/ω)))
  4. ω is its angular frequency: the number of complete oscillations f (t) makes in a time interval of length 2π; that is, the number of radians per unit time;
  5. v=ω/2π 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;
  6. P=2π/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(tτ)) .

 


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