Question

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x. (1 point)
f(x) = the quantity x minus seven divided by the quantity x plus three. and g(x) = quantity negative three x minus seven divided by quantity x minus one.

Answer 1

1) Functions given:

            x - 7
f(x) = ----------
           x + 3

            - 3x - 7
g(x) = ------------
               x - 1

                                                      
2)

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

3)

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

So we have proved that f[g(x)]=g[f(x)]=x