Question 177049
Your problem statement is a little unclear! Did you mean to write:
Verify algebraically that {{{f(f^(-1)(x)) = x}}} for {{{f(x) = 1/(2x+4)}}}
I'll assume that you did and we'll proceed.
First, we find {{{f^(-1)(x)}}}
Start with the given function, but replace the {{{f(x)}}}with {{{y}}}:
{{{y = (1/(2x+4))}}} Now exchange the x and the y.
{{{x = 1/(2y+4)}}} Next, solve this for y. Multiply both sides by (2y+4)
{{{x(2y+4) = 1}}} Divide both sides by x.
{{{2y+4 = 1/x}}} Subtract 4 from both sides.
{{{2y = (1/x)-4}}} Simplify the right side.
{{{2y = (1-4x)/x}}} Now divide both sides by 2.
{{{y = (1-4x)/2x}}} Finally, replace the {{{y}}} with {{{f^(-1)(x)}}}
{{{f^(-1)(x) = ((1-4x)/2x)}}}
Now you can find {{{f(f^(-1)(x))}}}
{{{f(f^(-1)(x)) = f((1-4x)/2x)}}} but {{{f(x) = 1/(2x+4)}}}, so...
{{{f((1-4x)/2x) = 1/((cross(2)((1-4x)/cross(2)x)+4))}}} Simplifying this...
{{{f((1-4x)/2x) = 1/((1cross(-4x)+cross(4x))/x)}}}
{{{f((1-4x)/2x) = 1/(1/x)}}} so...
{{{f(1-4x)/2x = x}}} and so
{{{f(f^(-1)(x)) = x}}} for {{{f(x) = 1/(2x+4)}}}