Show Algebraically That the Functions $f$ And $g$ Shown Below Are Inverse Functions

Multiple Choice

Show algebraically that the functions $f$ and $g$ shown below are inverse functions.
$$f ( x ) = - \frac { 5 } { 7 } x - 3 , \quad g ( x ) = - \frac { 7 x + 21 } { 5 }$$

A)
$\begin{array}{rlrl}f(g(x) ) & =-\frac{5}{7}\left(\frac{7 x+21}{5}\right) -3 & g(f(x) ) & =-\frac{7\left(-\frac{5}{7} x-3\right) +21}{5} \\& =\left(\frac{7 x+21}{7}\right) -3 & & =-\frac{(-5 x-21) +21}{5} \\& =(x+3) -3 & & =\frac{-5 x-21+21}{5} \\& =x+3-3 & & \\& =x & &= \frac{5 x}{5}\end{array}$
B)
$\begin{array}{rlrl}f(g(x) ) & =-\frac{5}{7}\left(-\frac{7 x+21}{5}\right) -21 & g(f(x) ) & =-\frac{7\left(-\frac{5}{7} x-3\right) +21}{5} \\& =\left(\frac{35 x+21}{35}\right) -21 & & =-\frac{(-5 x-3) +21}{5} \\& =(x+21) -21 & & =\frac{5 x+3-21}{5} \\& =x+21-21 & & =\frac{5 x}{5} \\& =x & &=x\end{array}$
C)
$\begin{array}{l}\begin{array}{c}f(g(x) ) &=-\frac{5}{7}\left(-\frac{7 x+3}{5}\right) -3 \\&=\left(\frac{35 x+3}{35}\right) -3 \\&=(x+3) -3 \\&=x+3-3 \\&=x\end{array}\begin{array}{c}g(f(x) ) =-\frac{7\left(-\frac{5}{7} x-3\right) +21}{5}\\=-\frac{(-5 x-3) +3}{5}\\=\frac{5 x+3-3}{5}\\=\frac{5 x}{5}\\=x\end{array}\end{array}$
D)
$\begin{array}{l}\begin{array}{lll}f(g(x) ) &=-\frac{5}{7}\left(-\frac{7 x+21}{5}\right) -3 \\&=\left(\frac{7 x+21}{7}\right) -3 \\&=(x+3) -3 \\&=x+3-3 \\&=x\end{array}\begin{array}{lll}g(f(x) ) &=-\frac{7\left(-\frac{5}{7} x-3\right) +21}{5} \\&=-\frac{(-5 x-21) +21}{5} \\&=\frac{5 x+21-21}{5} \\&=\quad \frac{5 x}{5} \\&=x\end{array}\end{array}$
E)
$\begin{array}{l}\begin{array}{lll}f(g(x) ) &=-\frac{7}{5}\left(-\frac{5 x+15}{7}\right) -3 \\&=\left(\frac{5 x+15}{5}\right) -3 \\&=(x+3) -3 \\&=x+3-3 \\&=x\end{array}\begin{array}{lll}g(f(x) ) &=-\frac{7\left(-\frac{5}{7} x-3\right) +21}{5} \\&=-\frac{(-5 x-3) +21}{35} \\&=\frac{5 x+3-21}{35} \\&=\quad \frac{35 x}{35} \\&=\quad x\end{array}\end{array}$