Answer:
[tex]f(x)=0.5\cdot (0.2)^x[/tex]
Step-by-step explanation:
Given the table
[tex]\begin{array}{cc}x&f(x)\\ \\-2&12.5\\ -1&2.5\\0&0.5\\1&0.1\\2&0.02\end{array}[/tex]
The exponential function can be written [tex]f(x)=a\cdot b^x.[/tex] To find [tex]a[/tex] and [tex]b,[/tex] substitute some values:
When [tex]x=0,\ f(x)=0.5,[/tex] then
[tex]0.5=a\cdot b^0\Rightarrow a=0.5\ [ b^0=1][/tex]
When [tex]x=1,\ f(x)=0.1,[/tex] then
[tex]0.1=0.5\cdot b^1\Rightarrow 0.5b=0.1,\ b=0.2[/tex]
Thus,
[tex]f(x)=0.5\cdot (0.2)^x[/tex]
Check remaining values:
[tex]f(-2)=0.5\cdot 0.02^{-2}=0.5\cdot 5^2=0.5\cdot 25=12.5\\ \\f(-1)=0.5\cdot 0.2^{-1}=0.5\cdot 5=2.5\\ \\f(2)=0.5\cdot 0.2^2=0.5\cdot 0.04=0.02[/tex]
