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


Question


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

Options

A)
f(x) = 2x + 3
B)
f(x) = 3x + 2
C)
f(x) = 4x + 1
D)
f(x) = 5x

The correct answer is C.

Explanation:

Choice C is correct. Because f is a linear function of x, the equation f(x) = mx + 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 \(\frac{8}{2}\), 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(x) = mx + b and solving for b. For example, using x = 1, f(x) = 5, and m = 4 yields 5 = 4(1) + b. Solving for b yields b = 1. Therefore, the equation defining the function f can be written in the form f(x) = 4x + 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 = 3, f(x) = 13. However, substituting x = 3 into the equation given in choice A gives f(3) = 2(3) + 3, or f(3) = 9, not 13. Similarly, substituting x = 3 into the equation given in choice B gives f(3) = 3(3) + 2, or f(3) = 11, not 13. Lastly, substituting x = 3 into the equation given in choice D gives f(3) = 5(3), or f(3) = 15, not 13. Therefore, the equations in choices A, B, and D cannot define f.


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