### Instruction: If $f(x) = \frac{3x+2}{5}$ what is the value of $\left[f^{-1}(4)\right]^{-1}$?### Response: This problem is asking me to find the inverse function of $f$ and then plug in $4$ and then take the reciprocal of that result.
To find the inverse function of $f$, I need to switch the roles of $x$ and $y$ and then solve for $y$.
So I start by writing $y = \frac{3x+2}{5}$ and then I swap $x$ and $y$ and get $x = \frac{3y+2}{5}$.
Now I need to isolate $y$.
To do that, I can multiply both sides by $5$ and get $5x = 3y+2$.
Then I can subtract $2$ from both sides and get $5x-2 = 3y$.
Then I can divide both sides by $3$ and get $y = \frac{5x-2}{3}$.
This is the inverse function of $f$.
To find $f^{-1}(4)$, I just need to plug in $4$ for $x$ in this formula.
So I get $f^{-1}(4) = \frac{5(4)-2}{3} = \frac{18}{3} = 6$.
To find $\left[f^{-1}(4)\right]^{-1}$, I just need to take the reciprocal of $6$.
So I get $\left[f^{-1}(4)\right]^{-1} = \frac{1}{6}$.