LibraryofFunctions.nb 1 A Library of Functions Any study of calculus must start with the study of functions. Functions are fundamental to mathematics. In its everyday use the word function conveys to us the idea that the knowledge of one fact leads to another. In mathematics, functions tell us how the knowledge of one number tells us another number. In this section will will review the behavior of the most common functions (powers, exponential, logarithms, and trigonometric). We will review the graphs of these functions, their domains and ranges, and the different methods of defining a function. Let's begin with a look at the graphs of the most common functions. Note that the domain and range of each function is listed below the graph. The Identity Function Identity Function, f x x x Range: All real numbers. The identity function is a special case of the class of functions known as linear functions. The linear functions are all functions whose graphs are straight lines. Linear functions can be defined in several ways. These definitions are listed below. Slope-intercept form A linear function in the form y = mx + b is in slope-intercept form, where m is the
LibraryofFunctions.nb slope of the line and b is the vertical intercept. Point-slope form A linear function in the form y - y 1 = m (x - x 1 ) is in point-slope form, where m is the slope of the line and the point x 1, y 1 is any point on the line. Standard form A linear equation in the form Ax + By = C is in standard form, where A, B, and C are integers. The identity function is the line through the origin (and hence has a vertical asymptote of 0) with a slope of 1. The Absolute Value Function The absolute value function is best described as a linear function that has "bounced off" the x-axis. Remember that the absolute value of an expression returns only positive values. To graph an absolute value function simply graph the function without the absolute value signs and reflect all parts of the graph below the x-axis above the x-axis. Absolute Value Function, f x x x Range: All real numbers greater than or equal to 0.
LibraryofFunctions.nb 3 The Square Function The square function is a special case of the quadratic functions. Quadratic functions are functions that can be defined by the equation f(x) = ax bx c, a 0. Quadratic functions can also be described as polynomial function of degree. For more information about polynomial functions see the section entitled "Polynomial and Rational Functions." The Square Function, f x x x Range: All real numbers greater than or equal to 0. The Cube Function The cube function is a special case of degree three polynomials which have the form f(x) = ax 3 bx cx d, a 0. Again, you can check out the section entitled "Polynomial and Rational Functions" for more about about these type of functions. The cube function is special because the b, c, and d in the general form are all 0.
LibraryofFunctions.nb The Cube Function, f x x 3 x Range: All real numbers. The Square Root Function The Square Root Function, f x x x Domain: All real numbers greater than or equal to 0. Range: All real numbers greater than or equal to 0.
LibraryofFunctions.nb 5 The Cube Root Function The Cube Root Function, f x 3 x x Range: All real numbers. The Natural Logarithmic Function The Natural Log. Function, f x ln x x Domain: All real number greter than 0. Range: All real numbers.
LibraryofFunctions.nb 6 The Exponential Function to Base e The Exponential Function, f x e x x Range: All real numbers greater than 0. The Trigonometric Functions The trigonometric functions are very important in the study of mathematics because of their cyclical properties. See the section entitled "Trigonometric Functions" for more information. The following are the graphs of three trigonometric functions; f(x) = sin x, f(x) = cos x, and f(x) = tan x.
LibraryofFunctions.nb 7 The Sine Function The Sine Function, f x sin x 1 x 1 Range: All real numbers between -1 and 1, inclusive. The Cosine Function The Cosine Function, f x cos x 1 x 1 Range: All real numbers between -1 and 1, inclusive.
LibraryofFunctions.nb 8 The Tangent Function The Tangent Function, f x tan x 3 3 x Domain: All real numbers not equal to odd multiples of /. Range: All real numbers Worksheet 1a