For each pair of functions $f$ and $g$ below, find $f(g(x))$ and $g(f(x))$.
Then, determine whether $f$ and $g$ are inverses of each other.
Simplify your answers as much as possible.
(Assume that your expressions are defined for all $x$ in the domain of the composition.
You do not have to indicate the domain.)
(a) $f(x)=-2 x$
(b) $f(x)=2 x+7$
$g(x)=-\frac{x}{2}$
$g(x)=\frac{x-7}{2}$
$f(g(x))=\square$
$f(g(x))=\square$
$g(f(x))=\square$
\[
g(f(x))=\square
\]
$f$ and $g$ are inverses of each other
$f$ and $g$ are inverses of each other
$f$ and $g$ are not inverses of each other
$f$ and $g$ are not inverses of each other
Check
2023

Step 1 ：\(f(x) = -2x\) and \(g(x) = -\frac{x}{2}\)

Step 2 ：Substitute \(g(x)\) into \(f(x)\) to find \(f(g(x))\)

Step 3 ：\(f(g(x)) = f(-\frac{x}{2}) = -2(-\frac{x}{2}) = x\)

Step 4 ：Substitute \(f(x)\) into \(g(x)\) to find \(g(f(x))\)

Step 5 ：\(g(f(x)) = g(-2x) = -\frac{-2x}{2} = x\)

Step 6 ：Since \(f(g(x)) = x\) and \(g(f(x)) = x\), \(f\) and \(g\) are inverses of each other

Step 7 ：\(f(x) = 2x + 7\) and \(g(x) = \frac{x-7}{2}\)

Step 8 ：Substitute \(g(x)\) into \(f(x)\) to find \(f(g(x))\)

Step 9 ：\(f(g(x)) = f(\frac{x-7}{2}) = 2(\frac{x-7}{2}) + 7 = x\)

Step 10 ：Substitute \(f(x)\) into \(g(x)\) to find \(g(f(x))\)

Step 11 ：\(g(f(x)) = g(2x + 7) = \frac{2x + 7 - 7}{2} = x\)

Step 12 ：Since \(f(g(x)) = x\) and \(g(f(x)) = x\), \(f\) and \(g\) are inverses of each other

Step 13 ：\boxed{\text{For both pairs of functions, } f \text{ and } g \text{ are inverses of each other}}