Answer: [tex]f(x)=2(0.5)^x[/tex]
Step-by-step explanation:
Given, the exponential function [tex]y = ab^x[/tex] for a graph that includes (–3, 16) and (–1, 4).
On putting these points, we will have
[tex]f(-3)=16=ab^{-3}\\\\ f(-1)=4=ab^{-1}[/tex]
Now,
[tex]\dfrac{f(-3)}{f(-1)}=\dfrac{16}{4}=\dfrac{ab^{-3}}{ab^{-1}}\\\\\Rightarrow\ \dfrac{4}{1}=\dfrac{b^{-3}}{b^{-1}}\\\\\Rightarrow\ \dfrac{4}{1}=\dfrac{b}{b^3}=\dfrac{1}{b^2}\\\\\Rightarrow\ b^2=\dfrac{1}{4}\\\\\Rightarrow\ b=\pm\dfrac{1}{2}=\pm0.5[/tex]
since the multiplicative factor cannot be negative, so b= 0.5.
At b= 0.5
[tex]4=a(0.5)^{-1}\\\\\Rightarrow\ 4=a(\dfrac{1}{2})^{-1}\\\\\Rightarrow\ 4=a(2)\\\\\Rightarrow \ a=2[/tex]
So, the required function is [tex]f(x)=2(0.5)^x[/tex].
