Answer:
The required function is [tex]f(x)=\frac{1}{2}\left(\frac{1}{4}\right)^x[/tex].
Step-by-step explanation:
The general exponential function is
[tex]f(x)=ab^x[/tex] .... (1)
where, a is the initial value and b is growth factor.
From the given table it is clear that the function passes through the points (0,0.5) and (-1,2). It means the equation of function must be satisfied by the points (0,0.5) and (-1,2).
Substitute f(x)=0.5 and x=0 in equation (1), to find the value of a.
[tex]0.5=ab^0[/tex]
[tex]0.5=a[/tex]
The value of a is 0.5.
Substitute a=0.5, f(x)=2 and x=-1 in equation (1), to find the value of b.
[tex]2=(0.5)b^(-1)[/tex]
[tex]2=\frac{0.5}{b}[/tex]
[tex]2b=0.5[/tex]
Divide both sides by 2.
[tex]b=\frac{0.5}{2}[/tex]
[tex]b=0.25[/tex]
The value of b is 0.25.
Substitute a=0.5 and b=0.25 in equation (1).
[tex]f(x)=0.5(0.25)^x[/tex]
[tex]f(x)=\frac{1}{2}(\frac{1}{4})^x[/tex]
Therefore the required function is [tex]f(x)=\frac{1}{2}\left(\frac{1}{4}\right)^x[/tex].
