Question

Will medal
Please help.
consider f(x) = 2x+1/x+3.
Find f^-1(x)

Answer 1

@dan815

Answer 2

So first you want to set your function as y =...

Answer 3

ok so 2y+1/y+3 = x

Answer 4

then

Answer 5

Alright now multiply x by y+3 to create a linear function.

Answer 6

What do you get with that.

Answer 7

xy+3x

Answer 8

\[x=\frac{2y+1}{y+3} \qquad xy +3x = 2y+1\] Do you see how this is working out?

Answer 9

xy+3x=2y+1, how did you do this

Answer 10

ohh did u cross muliply

Answer 11

\[(y+3) \cdot x= \frac{2y+1}{y+3}\cdot (y+3) \]

Answer 12

Yes.

Answer 13

ok, then

Answer 14

Then subtract -2y from both sides of the equation. \[xy-2y+3x=1\] And then youd subtract -3x from both sides of the equation. \[xy-2y=1-3x\]

Answer 15

Following?

Answer 16

why did -3x switch?

Answer 17

because we want to isolate all the terms with `y` to the left side of the equation, and 3x does not have a `y` so we send it to the other side.

In the process of solving for inverse functions, we switch x and y, then resolve for y.

Answer 18

ohhh ok, what to do next

Answer 19

Now we factor out a y from the left hand side of the equation. What do you end up with when you do that?

Answer 20

y(x-2)

Answer 21

Good.

Answer 22

now we divide both sides of the equation by (x-2) to solve for y. what is your equation then?

Answer 23

1-3x/x+2

Answer 24

careful with your signs. When you divide both sides by (x-2) it doesnt change to (x+2)

Answer 25

\[y(x-2) = 1-3x\qquad \implies \qquad y=\frac{1-3x}{x-2}\]

Answer 26

Thank you.

Answer 27

now all you have to do is rewrite y as f\(^{-1}\)(x) and this is your inverse function.