I seriously am having trouble with this problem! What is f-1(x). f(x)=(2x-3)/(x+1) The answer is f^-1(x)=(x+3)/(2-x) but how do I get that answer? My personal answer was (-(x+3)…

Question

I seriously am having trouble with this problem! What is f-1(x).
`f(x)=(2x-3)/(x+1)`

The answer is
`f^-1(x)=(x+3)/(2-x)` but how do I get that answer? My personal answer was
`(-(x+3))/(x-2)`

Please help me and explain how you came up with the actual answer. I greatly appreciate it and thank you!

Answer 1

f-1(x) = (x + 3)/ (2 - x) or -(x + 3 / (x - 2).

Explanation:

Let y = (2x - 3)/(x + 1)

We find x in terms of y:

Cross multiply:

y(x + 1) = 2x - 3

xy + y = 2x - 3

y + 3 = 2x - xy

x(2 - y) = y + 3

x = (y + 3) / (2 - y)

Now replace x by f-1(x) and y by x, we get:

f-1(x) = (x + 3)/ (2 - x).

Your answer was correct. You found the same result written in a different form.

If we multiply the above by -1 / -1 we get

-(x + 3) / -(2 - x)

= -(x + 3) / (x - 2).

Answer 2

see explanation

Explanation:

let y = f(x) and rearrange making x the subject, that is

y =
`(2x-3)/(x+1)` ← multiply both sides by (x + 1)

y(x + 1) = 2x - 3 ← distribute left side

xy + y = 2x - 3 ( subtract y from both sides )

xy = 2x - 3 - y ( subtract 2x from both sides )

xy - 2x = - 3 - y ← factor out x from each term on the left side

x(y - 2) = - 3 - y ← divide both sides by y - 2

x =
`(-3-y)/(y-2)` factor out - 1 on numerator and denominator

x =
`(-(3+y))/(-(2-y))`

Change y back into terms of x, thus

`f^(-1)`(x) =
`(3+x)/(2-x)` =
`(x+3)/(2-x)`