Question

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

f(x) =(x-7/x+3) and g(x) = (-3x-7/x-1)

Answer 1

f(g(x)) = x and g(f(x)) = x.

f(x) =(x-7/x+3) and g(x) = (-3x-7/x-1)

for f(g(x)), substitute the x's in x-7/x+3 by (-3x-7/x-1) and simplify to get x
similarly
for g(f(x)), substitute the x's in (-3x-7/x-1) by (x-7/x+3) and simplify to get x

Answer 2

yes, thats what the problem is asking for

Answer 3

\[f(x)=\frac{x-7}{x+3}\]\[g(x)=\frac{3x+7}{1-x}\](I moved a negative sign around in g(x) for simplicity
\[f(g(x))=\frac{\frac{3x+7}{1-x}-7}{\frac{3x+7}{1-x}+3}=\frac{\frac{3x+7-7(1-x)}{1-x}}{\frac{3x+7+3(1-x)}{1-x}}\]
\[=\frac{3x+7-7+7x}{3x+7+3-3x}=\frac{10x}{10}=x\]

Similarly for g(f(x))
\[g(f(x))=\frac{3\frac{x-7}{x+3}+7}{1-\frac{x-7}{x+3}}=\frac{\frac{3x-21+7(x+3)}{x+3}}{\frac{x+3-x+7}{x+3}}\]\[=\frac{3x-21+7x+21}{10}=\frac{10x}{10}=x\]