Question

Let f(x) = 8+sqrt(5 x-5) . Find f^(-1)(x)

Answer 1

y = 8 + sqrt(5x-5)
x = 8 + sqrt(5y-5)
x = 13sqrt(5y)
x = 13^2 * 5y^2
x = 65x^4

Can someone check my steps?

Answer 2

I think I did something wrong...

Answer 3

@iambatman @Hero

Answer 4

\[f(x) = 8+\sqrt{5x-5}\]
\[y = 8+\sqrt{5x-5}\]
\[x = 8+\sqrt{5y-5}\]
\[x - 8 = \sqrt{5y-5}\]
\[(x-8)^2 = (\sqrt{5y-5})^2\]
\[(x-8)^2 = 5y-5\]
\[(x-8)^2+5 = 5y\]
\[y = \frac{ (x-8)^2+5 }{ 5 }\]
\[y = \frac{ 1 }{ 5 }(x-8)^2 \implies f ^{-1}(x) = \frac{ 1 }{ 5 }(x-8)^2\]

Answer 5

@jtryon here are some steps that I used to achieve this.

To find the inverse:

Replace f(x) with y
Switch x's and y's, so put x where y is and x where y is.
Solve for y
Replace y with f^-1(x)

Answer 6

yeah, I remember you helping another user with inverse functions so I was trying to follow the steps

Answer 7

I gave you a step by step, so you can see how the process is, but next time you have to do this :P

Answer 8

I did remember to replace f(x) with y and switch x with y in the function

Answer 9

I think you got all that right, it was the algebra that gave you the trouble.

Answer 10

Yeah, unfortunately I'm still rusty on the algebra and I really need to polish that up

Answer 11

I have a bad habit of making my own algebra rules and breaking some

Answer 12

Oh I see you switched the x's and y's but you were trying to simplify it or something, but your y disappeared lol from there, but in actuality you're suppose to solve for y.

Answer 13

No worries :P, keep practicing!

Answer 14

thanks :P

Answer 15

I will use your example to study for my final

Answer 16

Good luck friend! ^.^

Answer 17

@iambatman, where did you get 1/5 for the last step in the algebra?

Answer 18

Oh I see you found a mistake ;), nice job, at the end it should be

\[f ^{-1}(x) = \frac{ 1 }{ 5 }((x-8)^2+5)\]

Answer 19

That's your final answer.

Answer 20

I'm still trying to figure out what step you did to get 1/5

Answer 21

\[f ^{-1}(x) = \frac{ (x-8)^2+5 }{ 5 }\]

You could just leave it as it is, I just factored the 1/5 out, you see dividing by 5 = 1/5.

Answer 22

I'm good all the way up to that point

Answer 23

Oh so you factored out a 1/5

Answer 24

Yeah it doesn't matter though, the final answer I wrote is wrong, because I forgot to add the +5, but yeah you could leave it as I showed you just now.

Answer 25

Hmm, it didn't take in that form so let me try with the 1/5 factored out

Answer 26

Oh maybe you have to put it as

\[f ^{-1}(x) = \frac{ 1 }{ 5 }(x^2-16x+69)\]

Answer 27

OK, that is what they wanted

Answer 28

I put it in with the 1/5((x-8)^2+5)

Answer 29

Did that work? That's what happens when you're dealing with computers, it's never specific enough.

Answer 30

Yeah, that answer with the 1/5 factored out did work.

Answer 31

Alright cool, glad that worked :P

Answer 32

Everything make sense now?

Answer 33

I'll tag you if I get stuck on something else... have a few more problems to work through. It does so you were multiplying by the reciprocal of 5 which is 1/5