Select the correct answer.
Exponential function [tex]$f$[/tex] is represented by the table.
| x | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|
| [tex]$f(x)$[/tex] | 78 | 24 | 6 | 0 | -2 |
Function [tex]$g$[/tex] 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 and increasing on the interval.
C. Both functions are positive and decreasing on the interval.
D. Both functions are positive on the interval, but one function is increasing while the other is decreasing.
Answer (1)
Analyze function f ( x ) from the table and determine it is positive and decreasing on ( 0 , 2 ) .
Determine the equation of function g ( x ) using the points ( 0 , 15 ) and ( 2 , 0 ) , resulting in g ( x ) = 15 − 7.5 x .
Analyze function g ( x ) and determine it is positive and decreasing on ( 0 , 2 ) .
Conclude that both functions are positive and decreasing on the interval ( 0 , 2 ) . The answer is C.
Explanation
Understanding the Problem We are given a table representing an exponential function f ( x ) and two points that an exponential function g ( x ) passes through. We need to compare the behavior of these two functions on the interval ( 0 , 2 ) .
Analyzing f(x) First, let's analyze the function f ( x ) using the given table:
x
-1
0
1
2
3
f ( x )
78
24
6
0
-2
On the interval ( 0 , 2 ) , we have the following values:
f ( 0 ) = 24
f ( 1 ) = 6
f ( 2 ) = 0
Since the values of f ( x ) are decreasing from 24 to 6 to 0, the function f ( x ) is positive and decreasing on the interval ( 0 , 2 ) .
Finding g(x) Next, let's find the equation of the exponential function g ( x ) . We are given two points ( 0 , 15 ) and ( 2 , 0 ) . Since the function is exponential, let's assume it has the form g ( x ) = a + b x . Substituting the given points, we get:
g ( 0 ) = a + b ( 0 ) = 15 , so a = 15 .
g ( 2 ) = a + b ( 2 ) = 0 , so 15 + 2 b = 0 , which means 2 b = − 15 , and b = − 7.5 .
Thus, the function g ( x ) is g ( x ) = 15 − 7.5 x .
Analyzing g(x) Now, let's analyze the behavior of g ( x ) on the interval ( 0 , 2 ) .
g ( 0 ) = 15 − 7.5 ( 0 ) = 15
g ( 1 ) = 15 − 7.5 ( 1 ) = 7.5
g ( 2 ) = 15 − 7.5 ( 2 ) = 0
Since the values of g ( x ) are decreasing from 15 to 7.5 to 0, the function g ( x ) is positive and decreasing on the interval ( 0 , 2 ) .
Conclusion Comparing the behaviors of f ( x ) and g ( x ) on the interval ( 0 , 2 ) , we found that both functions are positive and decreasing. Therefore, the correct statement is C.
Examples
Understanding the behavior of exponential functions is crucial in various real-world applications. For instance, in finance, exponential decay models the depreciation of an asset over time. Similarly, in biology, it can represent the decay of a drug concentration in the bloodstream. By analyzing the rate and direction of change, we can make informed decisions about investments, medical treatments, and other critical processes. In this case, we analyzed two functions, f ( x ) and g ( x ) , and determined that both are positive and decreasing on the interval ( 0 , 2 ) . This means that the values of both functions are diminishing as x increases from 0 to 2.
