Questions: Which of the following equations represents function f if f(14)=-40 and f(34)=-30 ? (A) f(x)=1/2 x-68 (B) f(x)=1/2 x-47 (C) f(x)=2 x-68 (D) f(x)=2 x-47

Solution Steps

We are given two points on the function \( f \): \( f(14) = -40 \) and \( f(34) = -30 \). We need to determine which of the given equations represents this function.

The slope \( m \) of a linear function can be calculated using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the given points \((14, -40)\) and \((34, -30)\):

\[ m = \frac{-30 - (-40)}{34 - 14} = \frac{10}{20} = \frac{1}{2} \]

Now that we know the slope is \(\frac{1}{2}\), we can eliminate options (C) and (D) because they have a slope of 2. We are left with options (A) and (B).

We need to check which of the remaining options satisfies both points. Let's test option (A) first:

• For \( x = 14 \):

\[ f(14) = \frac{1}{2} \times 14 - 68 = 7 - 68 = -61 \]

This does not match \( f(14) = -40 \), so option (A) is incorrect.

• For \( x = 14 \):

\[ f(14) = \frac{1}{2} \times 14 - 47 = 7 - 47 = -40 \]

• For \( x = 34 \):

\[ f(34) = \frac{1}{2} \times 34 - 47 = 17 - 47 = -30 \]

Both points satisfy the equation, so option (B) is correct.

Final Answer