Question

Find the Inverse of the Function f (x) = 3x-5/2x+1
How could you show the function is indeed the inverse?

Answer 1

the best way to show that the function has an inverse is to find it

Answer 2

the method to use, at least one method, is to take
\[y=\frac{3x-5}{2x+1}\] then switch \(x\) and \(y\) because that is what the inverse does, and get
\[x=\frac{3y-5}{2y+1}\] and then solve for \(y\)
it is a bunch of algebra, i can work through it with you if you like

Answer 3

yes please, how would you solve for y?

Answer 4

first lets get rid of the denominator by multiplying both sides by \(2y+1\) to get
\[x(2y+1)=3y-5\]

Answer 5

the remove parentheses using distributive law to get
\[2xy+x=3y-5\]

Answer 6

then put all the terms with \(y\) on one side, and all the terms without \(y\) on the other using addition and subtraction as needed to get
\[3y-2xy=x+5\]

Answer 7

factor out the \(y\) to get
\[(3-2x)y=x+5\]

Answer 8

and finally divide both sides by \(3-2x\) to get \(y\) by itself
you get \[y=\frac{x+5}{3-2x}\] unless i made an algebra mistake
check it

Answer 9

No mistake, but how would you show that the function is indeed the inverse? what would i have to do?

Answer 10

you would have to write
\[f^{-1}(x)=\frac{x+5}{3-2x}\] then carefully compute
\[f\circ f^{-1}(x)=f(f^{-1}(x))\] and see that the result is \(x\)

Answer 11

i see now, thanks!

Answer 12

well it is not trivial to compute the composition, but when you do there should be a whole raft of cancellation