Copyright © Cengage Learning. All rights reserved. 3.1 Graphs of Functions Copyright © Cengage Learning. All rights reserved.

Basic or Parent Functions There are six common functions that occur frequently in algebra. These graphs are sometimes referred to as basic functions or parent functions and can be sketched by plotting points. These common functions are the building blocks for graphing more complicated functions in algebra.

List of Common Functions The identify function f (x) = x pairs each real number with itself. See Figure (a). The squaring function f (x) = x2 pairs each real number with its square. See Figure (b).

List of Common Functions The cubing function f (x) = x3 pairs each real number with its cube. See Figure (c). The absolute value function f (x) = |x| pairs each real number with its absolute value. See Figure (d).

List of Common Functions The square root function f (x) = pairs each real number with its principal square root. See Figure (e). The cube root function f (x) = pairs each real number with its cube root. See Figure (f).

Example 1 Graph the function f (x) = –2|x| + 3

Example 2 Graph the function

Example 3  

The Vertical Line Test If every vertical line that intersects a graph does so exactly once, every number x determines exactly one value of y, and the graph represents a function.

Example 4 Determine which of the following graphs represent functions. We will use the vertical line test by drawing several vertical lines through each graph. If every vertical line that intersects the graph does so exactly once, the graph represents a function. Otherwise, the graph does not represent a function.

Determine Function Values In many applications that involve functions, we may not be given the equation of the function but will instead have only data. It is important that we learn how to read the value of a function from the graph of that function. Remember, the value of the function is y. It is not a point; it is a y-value.

Example 5 Refer to the graph of function f(x) shown in Figure. Find f (−3) Find f (1) Find the value of x for which f (x) = 17

The Domain and Range Both the domain and the range of a function can be identified by viewing the graph of the function. The inputs or x-values that correspond to points on the graph of the function can be identified on the x-axis and used to state the domain of the function. The outputs or f(x) values that correspond to points on the graph of the function can be identified on the y-axis and used to state the range of the function.

Example 6 Use the graph of each function to determine its domain and range.