Question 178280
Verify that {{{f^(-1)(f(x)) = x}}} and {{{f(f^(-1)(x)) = x}}} for:
{{{f(x) = x/(x+1)}}} for{{{x>-1}}}
First, you'll need to find {{{f^(-1)(x)}}}
Start with the given function:
{{{f(x) = x/(x+1)}}} Substitute {{{y}}} for {{{f(x)}}}
{{{y = x/(x+1)}}} Now exchange x and y.
{{{x = y/(y+1)}}} Next, solve this for y. Multiply both sides by {{{(y+1)}}}
{{{x(y+1) = y}}} Expand the left side.
{{{xy+x = y}}} Subtract xy from both sides.
{{{x = y-xy}}} Factor the y from the right side.
{{{x = y(1-x)}}} Divide both sides by {{{(1-x)}}}
{{{x/(1-x) = y}}} Now replace the {{{y}}} with {{{f^(-1)(x)}}}
{{{highlight(f^(-1)(x) = x/(1-x))}}}
Now you are in a position to find:
{{{f(f^(-1)(x))}}} Substitute {{{f^(-1)(x) = x/(1-x)}}} recalling that {{{f(x) = x/(x+1)}}}
{{{f(x/(1-x)) = ((x/(1-x)))/((x/(1-x))+1)}}} Notice that where there is an x on the left side, it's replaced with {{{x/(x-1)}}} on the right side. Now we'll simplify the right side.
{{{f(f^(-1)(x)) = (x/(x-1))/(1/(x-1))}}} Invert and multiply.
{{{f(f^(-1)(x)) = (x/cross((x-1)))*(cross((x-1))/1)}}}
{{{highlight(f(f^(-1)(x)) = x)}}}
Do you think you can manage to show that {{{f^(-1)(f(x)) = x}}}?