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

Question

Accepted Answer

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.

Question some-v

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

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