# Thread: function inverse

1. ## function inverse

Can anyone find the inverse function of this:

(will you please show work!)

Thank you!

2. Hi fruitkate21,

We have $\displaystyle y=f(x)=\frac{10x-1}{2x+9}$ Now interchange $\displaystyle y$ and $\displaystyle x$ rearranging for $\displaystyle y$;

$\displaystyle x=\frac{10y-1}{2y+9}$

$\displaystyle \implies 2xy+9x=10y-1$

$\displaystyle \implies 10y-2xy=9x+1$

$\displaystyle \implies y(10-2x)=9x+1$

$\displaystyle \implies y=f^{-1}(x)=\frac{9x+1}{10-2x}$

3. Hello, fruitkate21!

Find the inverse function of: .$\displaystyle f(x) \;=\;\frac{10x-1}{2x+9}$
There are four basic steps . . .

[1] Replace $\displaystyle f(x)$ with $\displaystyle y\!:\;\;y \;=\;\frac{10x-1}{2x+9}$

[2] Interchange $\displaystyle x$'s and $\displaystyle y$'s: .$\displaystyle x \;=\;\frac{10y - 1}{2y + 9}$

[3] Solve for $\displaystyle y\!:$

. . .$\displaystyle x(2y+9) \:=\:10y - 1$

. . . $\displaystyle 2xy + 9x \;=\;10y - 1$

. . $\displaystyle 2xy - 10y \;=\;\text{-}9x - 1$

. . .$\displaystyle 2(x-5)y \;=\;\text{-}(9x + 1)$

. . . . . . . .$\displaystyle y \;=\;\frac{\text{-}(9x+1)}{2(x-5)}$

[4] Replace $\displaystyle y$ with $\displaystyle f^{\text{-}1}(x)\!:\;\;\boxed{f^{\text{-}1}(x) \;=\;\frac{1 + 9x}{2(5-x)}}$

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

If you don't have to show your work,
. . there is a formula for this problem.

If $\displaystyle f(x) \:=\:\frac{{\color{blue}a}x + b}{cx + {\color{blue}d}}$ . . . then: .$\displaystyle f^{-1}(x) \;=\;\frac{{\color{blue}d}x \:{\color{red}-}\: b}{{\color{red}-}cx+ {\color{blue}a}}$

This is easily memorized:

. . (1) Switch the numbers on the main diagonal
. . . . .(upper-left and lower-right).

. . (2) Change the signs on the other diagonal
. . . . .(upper-right and lower-left).