Linear Functions Analysis: Which Functions are Positive When x > 2?

Choose the functions that fit the following description:

The function is positive in the domain $2 < x$.

a. $y=3x+4$

b. $y=2x-4$

c. $y=-2x+4$

d. $y=2$

e. $y=4x-8$

f. $y=5x-14$

To determine which functions are positive for the domain $x > 2$, we evaluate each function at $x = 2$ and use their properties:

• Function $y = 3x + 4$: At $x = 2$, $y = 3(2) + 4 = 10$, which is positive. As this is a linear function with a positive slope, it remains positive for $x > 2$.
• Function $y = 2x - 4$: At $x = 2$, $y = 2(2) - 4 = 0$. The function becomes positive for $x > 2$, as the slope is positive.
• Function $y = -2x + 4$: At $x = 2$, $y = -2(2) + 4 = 0$. The slope is negative, making it negative for $x > 2$.
• Function $y = 2$: This is a constant function with value 2, which is positive regardless of $x$.
• Function $y = 4x - 8$: At $x = 2$, $y = 4(2) - 8 = 0$. The positive slope indicates that it becomes positive for $x > 2$.
• Function $y = 5x - 14$: At $x = 2$, $y = 5(2) - 14 = -4$, which is negative, although it becomes positive for $x > 2$ (since the slope is positive, it crosses the x-axis soon after $x = 2$).

Based on this analysis, the correct answer is that the functions a, b, d, and e are positive for $x > 2$.