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)

Solution:

f(g(x)) and g(f(x)) are composite functions.

For f(g(x)) substitute g(x) for values of x in F(x)

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

Simplifying this further, we get

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

So we get,

-10x/-10 = x

Therefore, f(g(x)) = x is true.

For g(f(x)) substitute f(x) for values of x in g(x)

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

Simplifying this further, we get

g(f(x))= [(-3x +21 -7x-21)/(x+3)]/[(x+7 -x-3)/(x+3)]

-10x/-10 = x

Therefore, g(f(x)) = x is true.

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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)

Summary:

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).