Select the correct answer.
Exponential function [tex]$f$[/tex] is represented by the table.
Function [tex]$g$[/tex] is an exponential function passing through the points [tex]$(0,15)$[/tex] and [tex]$(2,0)$[/tex].
Which statement correctly compares the behavior of the two functions on the interval [tex]$(0,2)$[/tex]?
A. Both functions are positive on the interval, but one function is increasing while the other is decreasing.
B. Both functions are positive and decreasing on the interval.
C. Both functions are positive and increasing on the interval.
D. One function is positive on the interval, while the other is negative.
Answer (1)
Function f ( x ) is positive and decreasing on the interval ( 0 , 2 ) .
Assuming g ( x ) is a linear function, we find g ( x ) = − 7.5 x + 15 .
Function g ( x ) is also positive and decreasing on the interval ( 0 , 2 ) .
Both functions are positive and decreasing on the interval ( 0 , 2 ) , so the answer is B .
Explanation
Understanding the Problem The problem provides a table representing an exponential function f ( x ) and states that function g ( x ) is an exponential function passing through the points ( 0 , 15 ) and ( 2 , 0 ) . We need to compare the behavior of these two functions on the interval ( 0 , 2 ) .
Analyzing Function 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 0 as x goes from 0 to 2, f ( x ) is positive and decreasing on the interval ( 0 , 2 ) .
Analyzing Function g(x) Now, let's analyze the function g ( x ) . We are given that g ( x ) is an exponential function passing through the points ( 0 , 15 ) and ( 2 , 0 ) . However, an exponential function cannot pass through a point where y = 0 unless it is a horizontal line at y = 0 . Since g ( 0 ) = 15 , g ( x ) cannot be identically zero. Therefore, the problem statement that g ( x ) is an exponential function is incorrect. Instead, let's assume that g ( x ) is a linear function. A linear function can be defined as g ( x ) = a x + b . Using the points ( 0 , 15 ) and ( 2 , 0 ) , we can find the values of a and b .
Finding the Linear Function g(x) Using the point ( 0 , 15 ) , we have g ( 0 ) = a ( 0 ) + b = 15 , so b = 15 .
Using the point ( 2 , 0 ) , we have g ( 2 ) = a ( 2 ) + 15 = 0 , so 2 a = − 15 , and a = − 7.5 .
Thus, g ( x ) = − 7.5 x + 15 .
Analyzing the behavior of g(x) Now, let's analyze the behavior of g ( x ) = − 7.5 x + 15 on the interval ( 0 , 2 ) .
g ( 0 ) = − 7.5 ( 0 ) + 15 = 15
g ( 1 ) = − 7.5 ( 1 ) + 15 = 7.5
g ( 2 ) = − 7.5 ( 2 ) + 15 = 0
Since the values of g ( x ) are decreasing from 15 to 0 as x goes from 0 to 2, g ( x ) is positive and decreasing on the interval ( 0 , 2 ) .
Comparing the behavior of f(x) and g(x) Both functions f ( x ) and g ( x ) are positive and decreasing on the interval ( 0 , 2 ) .
Final Answer Therefore, the correct statement is B. Both functions are positive and decreasing on the interval.
Examples
Understanding the behavior of functions, like whether they are increasing or decreasing, is crucial in many real-world applications. For example, in economics, we might analyze a demand function to see how the quantity of a product demanded changes as its price increases. If the demand function is decreasing, it means that as the price goes up, the quantity demanded goes down. Similarly, in physics, we might study the velocity of an object as a function of time. If the velocity function is increasing, it means the object is accelerating. These concepts help us make informed decisions and predictions in various fields.
