Question

THE REWARD IS 20 PTS!!!

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

This is too complicated for me... please help... T-T

Answer 1

Explanation:

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

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

f(g(x))=(-3x-7-7x+7)/(-3x-7+3x-3)=-10x/-10=x

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

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

= -3x+ 21 -7x - 21 / 10 = x

Answer 2

Good use of parentheses in asking the question; fellow questioners take note.

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

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

It looks like we have some algebra ahead of us:

`f(g(x)) = f \left( (-3x - 7)/(x-1) \right) = ( \left( (-3x - 7)/(x-1) \right) - 7)/( \left( (-3x - 7)/(x-1) \right)+3)`

When we have a complicated fraction like that it's best if we multiply top and bottom by the inner denominators to simplify:

`f(g(x)) = ( \left( (-3x - 7)/(x-1) \right) - 7)/( \left( (-3x - 7)/(x-1) \right)+3) \cdot (x-1)/(x-1) = ( (-3x - 7) -7(x-1))/((-3x - 7)+3(x-1))`

`f(g(x)) = ( -3x -7 -7x+7)/(-3x - 7+3x-3) = (-10 x)/(-10) = x \quad\checkmark`

OK, now the other way.

`g(f(x)) = g \left( (x - 7)/(x+3) \right) = (-3 \left( (x - 7)/(x+3) \right) - 7)/( \left( (x - 7)/(x+3) \right)-1) \cdot (x+3)/(x+3)`

`g(f(x)) = (-3(x - 7) - 7(x+3))/( (x - 7)-(x+3)) = (-3x+21-7x-21)/(-10) = x \quad\checkmark`

Not so bad.