Precalculus: Functions

Types of Functions

In this section, we'll briefly cover a few of the most relevant and important classifications of functions.

Even and Odd Functions

Every function can either be classified as an even function, an odd function, or neither. Even functions have the characteristic that f (x) = f (- x) . They are symmetrical with respect to the y-axis. A line segment joining the points f (x) and f (- x) will be perfectly horizontal. Odd functions have the characteristic that f (x) = - f (- x) . They are symmetrical with respect to the origin. A line segment joining the points f (x) and - f (- x) always contains the origin. Many functions are neither even nor odd.

Some of the most common even functions are y = k , where k is a constant, y = x ² , and y = cos(x) . Some of the most common odd functions are y = x ³ and y = sin(x) . Some functions that are neither even nor odd include y = x - 2 , y = , and y = sin(x) + 1 .

Figure %: The function on the left is even; the function on the right is odd. Note the different types of symmetry.

Other Types of Functions

Among the types of functions that we'll study extensively are polynomial, logarithmic, exponential, and trigonometric functions. Before we study those, we'll take a look at some more general types of functions.

The inverse of a function is the relation in which the roles of the independent anddependent variable are reversed. Let f (x) = 2x . The inverse of f , f ^-1 (not to be confused with a negative exponent), equals . It is written like this: f ^-1(x) = . The inverse of a function can be found by switching the places of x and y in the formula of the function. The inverse of any function is a relation. Whether the inverse is a function depends on the original function f . If f is a one-to-one function, then its inverse is also a function. A one-to-one function is a function for which each element of the range corresponds to exactly one element of the domain. Therefore if a function is not a one-to- one function, its inverse is not a function. The horizontal line test shows us that if a horizontal line can be placed in a graph such that it intersects the graph of a function more than once, that function is not one-to-one, and its inverse is therefore not a function.

Inverse functions are important in solving equations. Sometimes the solution y to a function is known, but the input for that solution x is not known. In situations like these, the inverse of the function can be used to find x . We'll see more inverse functions later.

A piecewise function is a function which is defined by different rules depending on the value of the independent variable. The following piecewise function is graphed below: y = for x≤ 0 , y = x for 0 < x < 2 , and y = 2 for x≥2 .

Figure %: A piecewise function is governed by different rules depending on the value of the independent variable.

A periodic function is a function with the following characteristic: f (x) = f (x + c) , for all values of x , where c is some constant. This means that the values of f repeat themselves in a regular cycle. The most common periodic functions are the trigonometric functions, but others exist as well.