Section 3.3 Functions 185 Student Check 4 The number of cell phone subscribers in the United States, in millions, between 1985–2008 can be approximated by f (x) = 0.6501x2 - 3.0122x + 2.258, where x is the number of years after 1985. Use the function to determine the number of cell phone subscribers in the United States in the year 2008. (Source: http://www .infoplease.com) Troubleshooting Common Errors Some common errors associated with functions are shown. Objective 5 Examples A problem and an incorrect solution are given. Provide the correct solution and an explanation of the error. 5a. Determine if the relation {(-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4)} is a function. Incorrect Solution Correct Solution and Explanation The relation is not a function because -and 2 correspond same y-value. In a function, two x-values can correspond to the same y-value as long as each x-value corresponds to only 1 y-value. So, this relation is a function. beca 2 a pond to the sam 5b. Find f (3) if f (x) = 4x + 5. Incorrect Solution Correct Solution and Explanation f (3) = ( 4x + 5)( 3) f (3) = 12x + 15 f (3) does not mean multiplication. It means to find the output value for x = 3. f (3) = 4(3) + 5 f (3) = 12 + 5 f (3) = 17 Objective 5 ▶ Troubleshoot common errors. SUMMARY OF KEY CONCEPTS 1. A function is a relation in which each input can have only one output, that is, each x-value corresponds to exactly one y-value. The x-values of the function cannot occur more than once with different y-values if the relation is to be called a function. 2. The vertical line test can be used to determine if a graph represents a function. If any vertical line intersects a graph in more than one point, the graph does not represent a function. 3. Function notation, f (x), is another name for the output value y. The notation f (a) = b corresponds to the ordered pair (a, b). If the value of f (a) is unknown, we can evaluate the function at x = a to determine it. In an equation, we do this by substituting the number a for the variable in the function f. In a table or graph, we find the ordered pair in the table or on the graph whose x-value is a. The y-value of this ordered pair is f (a). 4. Functions occur in many real-world situations. It is important to understand from the problem what the input and output of the function represent. This enables us to find the requested information. ANSWERS TO STUDENT CHECKS Student Check 1 a. not a function b. function c. function d. function Student Check 2 a. function b. not a function Student Check 3 a. f (0) = 4 b. g (-5) = -29 c. c (2) = -6 d. f (-2) = 0 e. x=- 4 9 f. x = -1 g. x = 1 Student Check 4 about 276.88 million 3 x 5 2x
hendricks_intermediate_algebra_1e_ch1_3
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