# 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.

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