Functions, Limits, and Continuity: Functions

Functions, Limits, and Continuity Functions

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This graph is a line with y-intercept 0 and slope 2. The function f has the inverse g : R→R defined by g(x) = x/2.

The function denoted by f (x) = 2x may also be thought of as a function from the integers to the integers. It is not, however, a function from the real numbers to the integers, because when you put in a real number, you do not always get out an integer. For example, f (1/4) = 1/2, and 1/2 is not an integer.

(2) As an example of a more exotic function, let us construct a function from the set of names of the days in a week to the set of letters in the alphabet. We define the function g to take in the name of a day in the week and to give out the first letter in that name. For example, g(Wednesday) = W, and g(Sunday) = g(Saturday) = S. While this example shows how general the concept of a function is, for the rest of this course we will focus on functions from some subset of the real numbers to the real numbers.

Elementary Functions

In this section, we review the basic properties of the elementary functions studied in pre-calculus courses. These functions will be our main focus when applying the tools of differentiation and integration, so it is crucial to be familiar with them. The elementary functions include the linear, polynomial, rational, power, and trigonometric functions.

Linear Functions

We already saw one example of a linear function above, f (x) = 2x. A general linear function (so called because its graph is a line) has the form f (x) = ax + b, where a and b are real numbers. The number a is called the slope of f and indicates how steeply inclined is the graph of f. The number b is called the $y$-intercept of f and is equal to f (0), the value of the function when its graph intersects the vertical axis, or the y-axis. This is illustrated in the figure below:

Figure %: Plot of f (x) = ax + b and y-intercept at b

All linear functions are invertible. The inverse of f (x) = ax + b is the function g(x) = (1/a)x + (- b/a), which also happens to be linear. Check that g is indeed an inverse for f.