Question 73977
To find the inverse of a function, set the function equal to y , exchange the x's and the y, then solve for y. The given function is: {{{f(x) = (3+2x)/(2-5x)}}} Set the function = y'
{{{y = (3+2x)/(2-5x)}}} Exchange the x's for y.
{{{x = (3+2y)/(2-5y)}}} Now solve for y. Multiply both sides by (2-5y)
{{{x(2-5y) = 3+2y}}} Simplify the left side.
{{{2x-5xy = 3+2y}}} Add 5xy to both sides.
{{{2x = 3+5xy+2y}}} Subtract 3 from both sides.
{{{2x-3 = 5xy+2y}}} Factor yhe y on the right side.
{{{2x-3 = y(5x+2)}}} Divide both sides by (5x+2)
{{{(2x-3)/(5x+2) = y}}}, so...
{{{f^(-1)(x) = (2x-3)/(5x+2)}}}
Checking ths solution:
We want to show that:
{{{f(f^(-1)(x)) = x}}}
{{{f((2x-3)/(5x+2)) = (3+2((2x-3)/(5x+2)))/(2-5((2x-3)/(5x+2)))}}} Simplify the right side.
{{{f((2x-3)/(5x+2)) = ((3(5x+2)+2(2x-3))/(5x+2))/((2(5x+2)-5(2x-3))/(5x+2))}}} Continue to simplify.
{{{f((2x-3)/(5x+2)) = (3(5x+2)+2(2x-3))/(2(5x+2)-5(2x-3))}}}
{{{f((2x-3)/(5x+2)) = (15x+6+4x-6)/(10x+4-10x+15)}}}={{{19x/19 = x}}} Whew!!!
So, {{{f(f^(-1)(x)) = x}}}