Question

I seriously am having trouble with this problem! What is f-1(x). 7Bx%2B1%7D" id="TexFormula1" title="f(x)=\frac{2x-3}{x+1}" alt="f(x)=\frac{2x-3}{x+1}" align="absmiddle" class="latex-formula">
The answer is $f^-1(x)=\frac{x+3}{2-x}$ but how do I get that answer? My personal answer was $\frac{-(x+3)}{x-2}$

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

Answer 1

Answer:

see explanation

Step-by-step explanation:

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

y = $\frac{2x-3}{x+1}$ ← 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 = $\frac{-3-y}{y-2}$ factor out - 1 on numerator and denominator

x = $\frac{-(3+y)}{-(2-y)}$

Change y back into terms of x, thus

$f^{-1}$(x) = $\frac{3+x}{2-x}$ = $\frac{x+3}{2-x}$

Answer 2

Answer:

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

Step-by-step 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).