# Using data from 2009 through 2015, the percent of households with internet acces

Using data from 2009 through 2015, the percent of households with internet access in a certain region can be modeled by the function $f\left(x\right)=51.952{x}^{0.16}$ to the number of years after 2005.
a. Find the formula for the inverse of the function f.
b. If the inverse is ${f}^{-1}$, find ${f}^{-1}.\left(f\left(20\right)\right)$
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Step 1
Solve for function inverse
Step 2
$f\left(x\right)=51.952{x}^{0.160}$
Now $x=51.952{\left[f\left(x\right)\right]}^{0.160}$
$\frac{x}{51.952}={\left[f\left(x\right)\right]}^{0.160}$
${\left[\frac{x}{51.952}\right]}^{\frac{1}{0.160}}=f\left(x\right)$
Now ${f}^{-1}\left(x\right)={\left(\frac{x}{51.952}\right)}^{\frac{1}{0.160}}$
a) ${F}^{-1}\left(x\right)={\left(\frac{x}{51.952}\right)}^{\frac{1}{0.160}}$
b) ${F}^{-1}\left(F\left(20\right)\right)={F}^{-1}\left(51.952{\left(20\right)}^{0.160}\right)$
$={\left(\frac{51.952{\left(20\right)}^{0.160}}{51.952}\right)}^{\frac{1}{0.160}}$
$={\left[{\left(20\right)}^{0.160}\right]}^{\frac{1}{0.160}}$
${f}^{-1}\left(f\left(20\right)\right)=20$