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Algebra / Linear functions Difficulty: Easy
The figure presents a 2-column table with 3 rows of data. The heading for the first column is “x.” The heading for the second column is “f of x.” The three rows of data are as follows. Row 1. x, 1; f of x, 5. Row 2. x, 3; f of x, 13. Row 3. x, 5; f of x, 21.

Some values of the linear function f are shown in the table above. Which of the following defines f ?

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Explanation

Choice C is correct. Because f is a linear function of x, the equation f of x equals, m x plus b, where m and b are constants, can be used to define the relationship between x and f (x). In this equation, m represents the increase in the value of f (x) for every increase in the value of x by 1. From the table, it can be determined that the value of f (x) increases by 8 for every increase in the value of x by 2. In other words, for the function f the value of m is eight halves, or 4. The value of b can be found by substituting the values of x and f (x) from any row of the table and the value of m into the equation f of x equals, m x plus b and solving for b. For example, using x equals 1, f of x equals 5, and m equals 4 yields 5 equals, 4 times 1, plus b. Solving for b yields b equals 1. Therefore, the equation defining the function f can be written in the form f of x equals, 4 x plus 1.

Choices A, B, and D are incorrect. Any equation defining the linear function f must give values of f (x) for corresponding values of x, as shown in each row of the table. According to the table, if x equals 3, f of x equals 13. However, substituting x equals 3 into the equation given in choice A gives f of 3 equals, 2 times 3, plus 3, or f of 3 equals 9, not 13. Similarly, substituting x equals 3 into the equation given in choice B gives f of 3 equals, 3 times 3, plus 2, or f of 3 equals 11, not 13.

Lastly, substituting x equals 3 into the equation given in choice D gives f of 3 equals, 5 times 3, or f of 3 equals 15, not 13. Therefore, the equations in choices A, B, and D cannot define f.