To determine which function has the values of its range decrease as the values in its domain increase, we need to identify an exponential decay function. An exponential decay function is characterized by a base between 0 and 1, where the function generally takes the form [tex]\( f(x) = a^x \)[/tex] with [tex]\( 0 < a < 1 \)[/tex].
Let's analyze each of the given functions:
1. [tex]\( k(x) = \frac{1}{2}(3)^x \)[/tex]:
- This function can be rewritten as [tex]\( k(x) = \frac{1}{2} \cdot 3^x \)[/tex].
- The base [tex]\( 3 \)[/tex] is greater than 1, indicating this function is an exponential growth function, not decay.
2. [tex]\( g(x) = 3.5^x \)[/tex]:
- The base [tex]\( 3.5 \)[/tex] is greater than 1, indicating this is also an exponential growth function.
3. [tex]\( h(x) = 0.3^x \)[/tex]:
- The base [tex]\( 0.3 \)[/tex] is between 0 and 1.
- Therefore, this is an exponential decay function. As [tex]\( x \)[/tex] increases, [tex]\( 0.3^x \)[/tex] decreases.
4. [tex]\( f(x) = 3^x \)[/tex]:
- The base [tex]\( 3 \)[/tex] is greater than 1, indicating this is another exponential growth function.
Given these analyses, the function [tex]\( h(x) = 0.3^x \)[/tex] is the only one that decreases as the values in its domain increase, identifying it as an exponential decay function. Consequently, the correct answer is:
C. [tex]\( h(x) = 0.3^x \)[/tex]
