Answer
$a.$
$y=f^{-1}(x)=\dfrac{6-x}{4}$
$b.$
The relationships $f^{-1}(f(x))=f(f^{-1}(x))=x$ are verified.
Work Step by Step
$f(x)=6-4x$
$a.$
Substitute $f(x)$ by $y$:
$y=6-4x$
Solve the equation for $x$. Begin by taking $4x$ to the left side and $y$ to the right side:
$4x=6-y$
Take the $4$ to divide the right side:
$x=\dfrac{6-y}{4}$
Interchange $x$ and $y$ and write the inverse in $y=f^{-1}(x)$ form:
$y=f^{-1}(x)=\dfrac{6-x}{4}$
$b.$
Check the inverse found by evaluating $f(f^{-1}(x))$ and $f^{-1}(f(x))$:
$f(f^{-1}(x))=6-4\Big(\dfrac{6-x}{4}\Big)=6-6+x=x$
$f^{-1}(f(x))=\dfrac{6-(6-4x)}{4}=\dfrac{6-6+4x}{4}=\dfrac{4x}{4}=x$
The relationships $f^{-1}(f(x))=f(f^{-1}(x))=x$ are verified.
