Question 681109
Tell whether or not the functions are inverses of each other?

1)F(x)=6x-7, g(x)=(x-6)/7

and 

2) f(x)=3/(x+7), g(x)=(7x+3)/x

<br> If they are inverses, then their composite functions have to equal x <br>
That is f(g(x))=x and g(f(x))=x.

1. {{{f(g(x))=f((x-6)/7)}}}
You will substitute g(x) in for the x's in f(x).
{{{f(g(x))=6((x-6)/7)-7}}}
{{{f(g(x))=(6x-36)/7-7}}}
Get a common denominator...
{{{f(g(x))=(6x-36)/7-49/7}}}
{{{f(g(x))=(6x-85)/7}}}
since it did not simplify to x, they are not inverses of each other.

<br>2. f(g(x))=f((7x+3)/x)<br>
{{{f(g(x))=3/((7x+3)/x+7)}}}
Multiply the top and the bottom by x to clear the complex fraction.
{{{f(g(x))=3x/(x(7x+3)/x+7x)}}}
{{{f(g(x))=3x/(7x+3+7x)}}}
{{{f(g(x))=3x/(14x+3)}}}
This also doesn't simplify to x and therefore are not inverses.

******************************************************************
Just to clarify....this is an example where they ARE inverses.
3. {{{f(x)=3/(x-1)}}} and {{{g(x)=(3+x)/x}}}
{{{f(g(x))=f((3+x)/x)}}}
{{{f(g(x))=3/(((3+x)/x)-1)}}}
to clear the complex fraction, multiply the top and bottom by x
{{{f(g(x))=3x/(x(3+x)/x-x)}}}
{{{f(g(x))=3x/(3+x-x)}}}
{{{f(g(x))=3x/3}}}
{{{f(g(x))=x}}}
f(x) and g(x) ARE inverses in this case because f(g(x))=x.  
*Note...g(f(x))=x as well....give it a try.

Hope that helped.  Happy Calculating!!!!