# Math Help - inverse of a function

1. ## inverse of a function

Im trying to find the inverse of the function with working, see attached image

2. $f(x) = \frac{2 - 3x}{4x + 5}$

Say $f(x) = y$

$y = \frac{2 - 3x}{4x + 5}$

Thus :

$y(4x + 5) = 2 - 3x$

Expand :

$4xy + 5y = 2 - 3x$

Rearrange terms :

$4xy + 3x = 2 - 5y$

Factorize $x$ :

$x(4y + 3) = 2 - 5y$

Make x the subject by dividing :

$x = \frac{2 - 5y}{4y + 3}$

Thus the inverse function of $f(x)$ is $f(x)^{-1} = \frac{2 - 5x}{4x + 3}$

Does it make sense ?

3. Cheers Ray that is spot on

4. put $y=\frac{2-3x}{4x+5}$,flip $x$ and $y$ , $x=\frac{2-3y}{4y+5}$ and solve for $y$.
$x=\frac{2-3y}{4y+5}\Leftrightarrow x\left ( 4y+5 \right )$= $2-3y\Leftrightarrow 4xy+3y+5x$= $2\Leftrightarrow y\left ( 4x+3 \right )$= $2-5x\Leftrightarrow y=\frac{2-5x}{4x+3}$
and hence the inverse is $f^{-1}(x)=\frac{2-5x}{4x+3}$

5. Originally Posted by hunterage2000
Cheers Ray that is spot on
By the way, this trick can help you find answers easily so as to be able to solve this type of questions quicker : can you spot some kind of relation between the numbers in the function and the numbers in the inverse function ?

EDIT : woah Raoh now this is "compressed" maths, now I realize how much of a waste of paper my writing style would be in an exam . I detailed carefully each step because Hunterage asked for it though .

6. Originally Posted by Bacterius
By the way, this trick can help you find answers easily so as to be able to solve this type of questions quicker : can you spot some kind of relation between the numbers in the function and the numbers in the inverse function ?

EDIT : woah Raoh now this is "compressed" maths, now I realize how much of a waste of paper my writing style would be in an exam . I detailed carefully each step because Hunterage asked for it though .
Yes but I founds yours easier to follow. Bear in mind that $x=-\frac{5}{4}$ is not in the domain of f(x)