An exponential function is characterized by an initial value, and an exponential rate
The function f(x) is [tex]\mathbf{f(x) =0.5(0.2)^x}[/tex]
f(x) is an exponential function that passes through points (-2,12.5), (-1,2.5), (0,0.5), (1,0.1) and (2,0.2).
An exponential function is represented as:
[tex]\mathbf{y = ab^x}[/tex]
At point (0,0.5), we have:
[tex]\mathbf{0.5 = ab^0}[/tex]
This gives
[tex]\mathbf{0.5 = a \times 1}[/tex]
[tex]\mathbf{0.5 = a}[/tex]
Rewrite as:
[tex]\mathbf{a = 0.5 }[/tex]
At point (-1,2.5), we have:
[tex]\mathbf{2.5 = ab^{-1}}[/tex]
Substitute 0.5 for a
[tex]\mathbf{2.5 = 0.5b^{-1}}[/tex]
Divide both sides by 0.5
[tex]\mathbf{5 = b^{-1}}[/tex]
Take inverse of both sides
[tex]\mathbf{\frac 1{5} = b}[/tex]
Rewrite as:
[tex]\mathbf{b = \frac 1{5} }[/tex]
[tex]\mathbf{b = 0.2}[/tex]
So, we have:
[tex]\mathbf{b = 0.2}[/tex] and [tex]\mathbf{a = 0.5 }[/tex]
Substitute these values in [tex]\mathbf{y = ab^x}[/tex]
[tex]\mathbf{y =0.5(0.2)^x}[/tex]
Express as a function
[tex]\mathbf{f(x) =0.5(0.2)^x}[/tex]
Hence, the function f(x) is [tex]\mathbf{f(x) =0.5(0.2)^x}[/tex]
Read more about exponential functions at:
https://brainly.com/question/11487261
