please help!

\(\text{we find the inverse by setting }y=f(x)\\ \text{solving for }x \text{ in terms of }y \text{, and then substituting }y \to x\)

\(f(x)=5x-12\\ y=5x-12\\ x = \dfrac{y+12}{5}\\ f^{-1}(x) = \dfrac{x+12}{5}\)

\(f^{-1}(x) = f(x+1)\\ \dfrac{x+12}{5} = 5(x+1)-12\\ x+12 = 25(x+1)-60\\ 24x = 47\\ x = \dfrac{47}{24}\)

If f(x)=5x-12, find a value for x so that f^-1(x)=f(x+1).

\(\begin{array}{|rclcrcl|} \hline f\Big(f^-1(x)\Big) &=& x & | & \quad f^-1(x)&=& f(x+1) \\ f\Big(f(x+1)\Big) &=& x & | & \quad f(x+1) &=& 5(x+1) - 12\\ && & | & \quad &=& 5x - 7 \\ f\Big(5x - 7)\Big) &=& x \\ 5(5x-7)-12 &=& x \\ 25x-35-12 &=& x \\ 24x &=& 47 \\ \mathbf{x} & \mathbf{=} & \mathbf{\dfrac{47}{24}} \\ \hline \end{array}\)

First find f^-1(x).

\(x = 5f^{-1}(x)-12\\ f^{-1}(x) = \dfrac{x+12}{5}\).

Equate this to f(x+1), which is 5(x+1)-12.

\(\dfrac{x+12}{5}=5x-7\\ x+12 = 25x-35\\ 24x=47\\ x = \dfrac{47}{24}\)