Basic Classes of Functions

Linear Functions and Slope

Linear functions : functions which have the form f(x) = ax + b, where a and b are constants

Slope : the change in y for each unit change in x(this ratio is independent of the points chosen), measurement of both the steepness and the direction of a line

slope of the line

We conclude that the formula f(x) = ax + b tells us the slope, a, and the y-intercept, for this line.

slope-intercept form

Sometimes it is convenient to express a linear function in different ways. For example, suppose the graph of a linear function passes through the point (x1, y1) and the slope of the line is m.

point-slope equation

A vertical line is described by the equation x = k for some constant k. Since neither the slope-intercept form nor the point-slope form allows for vertical lines, we use the notation

where a, b are both not zero, to denote the standard form of a line.

 

 

Polynomials

A linear function is a special type of a more general class of functions: polynomials. A polynomial function is any function that can be written in the form

for some integer n ≥ 0 and constants an, an−1,…,a0, where a_n ≠ 0.

The value n is called the degree of the polynomial; the constant an is called the leading coefficient.

A polynomial of degree 0 is also called a constant function.

A polynomial function of degree 2 is called a quadratic function.

A polynomial function of degree 3 is called a cubic function.

 

※ Power Functions

A power function is any function of the form

where a and b are any real numbers. If the exponent is a positive integer, then the power function is a polynomial.

 

※ Behavior at Infinity

For some functions, the values of f(x) approach a finite number. ex) f(x)=1/x

The line y = 0 is a horizontal asymptote for the function.

For other functions, the values f(x) may not approach a finite number but instead may become larger for all values of x as they get larger. In that case, we say “f(x) approaches infinity as x approaches infinity,”

 

※ Zeros of Polynomial Functions

Another characteristic of the graph of a polynomial function is where it intersects the x-axis.

In the case of the linear function f(x) = mx + b, we see that the x-intercept is given by (−b/m, 0).

In the case of a quadratic function, finding the x-intercept(s) requires finding the zeros of a quadratic equation.

In the case of higher-degree polynomials, it may be more complicated to determine where the graph intersects the x-axis. In some instances, it is possible to find the x-intercepts by factoring the polynomial to find its zeros. In other cases, it is impossible to calculate the exact values of the x-intercepts. However, as we see later in the text, in cases such as this, we can use analytical tools to approximate (to a very high degree) where the x-intercepts are located.

 

※ Mathematical Models

A mathematical model is a method of simulating real-life situations with mathematical equations. Models are useful because they help predict future outcomes.

 

Algebraic Functions

Byallowing for quotients and fractional powers in polynomial functions, we create a larger class of functions. An algebraic function is one that involves addition, subtraction, multiplication, division, rational powers, and roots. Two types of algebraic functions are rational functions and root functions.

a rational function is any function of the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomials.

A root function is a power function of the form

where n is a positive integer greater 3 than one

 

Transcendental Functions

Some functions, however, cannot be described by basic algebraic operations. These functions are known as transcendental functions because they are said to “transcend,” or go beyond, algebra. The most common transcendental functions are trigonometric, exponential, and logarithmic functions.

 

Transformations of Functions

A shift, horizontally or vertically, is a type of transformation of a function. Other transformations include horizontal and vertical scalings, and reflections about the axes.