f(x) = (2x+3)/(4x-5)
So, I let f(x) = y and then I get y = (2x+3) / (4x-5) and then I figure I solve for x... But I am having difficulty in this, any suggestions? Thank you
I found
$\displaystyle \frac{2x+3}{4x-5}= \frac{1}{2(4x-5)}+\frac{1}{2}$
Now $\displaystyle y = \frac{1}{2(4x-5)}+\frac{1}{2}$
To find the inverse, swap y and x and solve for y.
$\displaystyle x = \frac{1}{2(4y-5)}+\frac{1}{2}$
$\displaystyle x-\frac{1}{2} = \frac{1}{2(4y-5)}$
$\displaystyle 4y-5 = \frac{1}{2(x-\frac{1}{2})}$
$\displaystyle 4y = \frac{1}{2(x-\frac{1}{2})}+5$
$\displaystyle y = \frac{1}{8(x-\frac{1}{2})}+\frac{5}{4}$
$\displaystyle y = \frac{1}{8x-4}+\frac{5}{4}$
to find inverse, i'm pretty sure you have to exchange the Y with the X, and then solve for Y.
original equation:
$\displaystyle y=\frac{2x+3}{4x-5}$
exchange variables:
$\displaystyle x=\frac{2y+3}{4y-5}$
solve for y:
$\displaystyle x(4y-5)=2y+3$ <----move the denominator on the right to the left
$\displaystyle 4xy-5x=2y+3$ <----factor
$\displaystyle 4xy-2y=5x+3$ <----move all the y's to one side and the single x to the other
$\displaystyle 2y(2x-1)=5x+3$ <----factor out the 2y from the left side
$\displaystyle 2y=\frac{5x+3}{2x-1}$ <----get the 2y by itself
$\displaystyle y=\frac{5x+3}{4x-2}$ <----get the y by itself
EDIT: I think my way is easier, pickslides...