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SOLUTION
[tex]\begin{gathered} f(x)=(\frac{1}{8})2^x \\ f^{-1}(x)=? \end{gathered}[/tex]
To determine the inverse function;
[tex]Solve\text{ the equation for x, then interchange x for y.}[/tex][tex]Let\text{ y=f\lparen x\rparen}[/tex][tex]\begin{gathered} y=\frac{1}{8}(2)^x \\ 8y=2^x \\ ln(8y)=ln2^x \\ ln(8y)=xln2 \\ x=\frac{ln(8y)}{ln2}\text{ ie y=}\frac{ln(8x)}{ln2} \\ \therefore f^{-1}(x)=\frac{ln(8x)}{ln(2)} \end{gathered}[/tex]
Now,
[tex]f^{-1}(\frac{1}{2})=\frac{ln(8\times\frac{1}{2})}{ln2}=\frac{ln4}{ln2}=2[/tex][tex]f^{-1}(8)=\frac{ln(8\times8)}{ln2}=\frac{ln(64)}{ln2}=6[/tex]
