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