Question

Select the correct answer.
Exponential function f is represented by the table.

x -1 0 1 2 3
f(x) 78 24 6 0 -2

Function g is an exponential function passing through the points (0,15) and (2,0).

Which statement correctly compares the behavior of the two functions on the interval (0, 2)?

A.
One function is positive on the interval, while the other is negative.
B.
Both functions are positive on the interval, but one function is increasing while the other is decreasing.
C.
Both functions are positive and decreasing on the interval.
D.
Both functions are positive and increasing on the interval.

Answer 1

To analyze the behavior of the functions \( f(x) \) and \( g(x) \) on the interval \( (0, 2) \), let's examine each function.

**Function \( f(x) \):**
From the table given, we know:
- \( f(0) = 24 \) (positive)
- \( f(1) = 6 \) (positive)
- \( f(2) = 0 \) (not positive; exactly 0)

Between \( x = 0 \) and \( x = 2 \), \( f(x) \) decreases from 24 to 0. However, it is still positive for \( x \) in \( (0, 2) \).

**Function \( g(x) \):**
We know that \( g(x) \) is an exponential function passing through the points \( (0, 15) \) and \( (2, 0) \).
At \( x = 0 \), \( g(0) = 15 \) (positive).
At \( x = 2 \), \( g(2) = 0 \) (not positive; exactly 0).
Since \( g(x) \) is an exponential function, it must also be decreasing on the interval \( (0, 2) \) as it passes from a positive value to 0.

**Comparing the two functions on the interval \( (0, 2) \):**
- Both \( f(x) \) and \( g(x) \) start positive at \( x = 0 \) and are decreasing towards 0 by \( x = 2 \).
- Thus, both functions are positive on \( (0, 2) \), but both are decreasing.

**Answer:** C. Both functions are positive and decreasing on the interval.