To determine which function will have a graph that decreases, we can analyze the nature of each function. The key characteristic we're looking at here is whether the exponential term in each function increases or decreases as [tex]\( x \)[/tex] increases.
1. [tex]\( f(x) = 1.25(0.3)^x + 7 \)[/tex]:
The exponential term here is [tex]\( (0.3)^x \)[/tex]. Notice that the base of the exponential term, 0.3, is less than 1. When the base of an exponential function is less than 1, the function will decrease as [tex]\( x \)[/tex] increases. Thus, this function will have a decreasing graph.
2. [tex]\( f(x) = 13(2.5)^x - 7 \)[/tex]:
The exponential term here is [tex]\( (2.5)^x \)[/tex]. The base of the exponential term, 2.5, is greater than 1. When the base of an exponential function is greater than 1, the function will increase as [tex]\( x \)[/tex] increases. Thus, this function will have an increasing graph.
3. [tex]\( f(x) = 7(3.5)^x - 0.25 \)[/tex]:
The exponential term here is [tex]\( (3.5)^x \)[/tex]. The base of the exponential term, 3.5, is greater than 1. Therefore, this function will also increase as [tex]\( x \)[/tex] increases, leading to an increasing graph.
4. [tex]\( f(x) = 0.1(7)^x - 2.3 \)[/tex]:
The exponential term here is [tex]\( (7)^x \)[/tex]. The base of the exponential term, 7, is greater than 1. Consequently, this function will increase as [tex]\( x \)[/tex] increases, resulting in an increasing graph.
In summary, only one of the functions listed has a base in its exponential term that is less than 1, and therefore, only this function's graph will decrease. That function is:
[tex]\[ f(x) = 1.25(0.3)^x + 7 \][/tex]
Thus, the answer is the first function in the list.
