Question

Does anyone know how to solve for the inverse of f(x) = -(1/(x-6)-4 then graph it?

Answer 1

I'm little bit confused by your ( )'s, is the function:\[\Large f(x)=-\frac{ 1 }{x-6 }-4\]

Answer 2

yeah, the sheet gave us a parent functions and told us the transformations and wanted us to rewrite the equation and then graph its unverse.

Answer 3

inverse*

Answer 4

You'll want to "switch and solve", meaning, write in y= form (just replace the f(x) with a y:

\[\Large y=-\frac{ 1 }{x-6 }-4\]

Then SWITCH the x and the y (anywhere you have a y, make it an x; anyhwere you have an x, make it a y):
\[\Large x=-\frac{ 1 }{y-6 }-4\]

Now, solve that^^ for y. That is your function \(\large y=f^{-1} (x)\)

Answer 5

I got to that stage, but i don't know what to do first. Bring the 4 over?

Answer 6

Yes, that sounds like a good start. :)

Answer 7

It does, but once i start breaking up the fractions it looks messy.

\[x+4 = -\frac{ 1 }{ y-6 }\]

then ill multiply by the denominator

\[(y-6) x+4 = -1 \]

and that just looks wrong to me

Answer 8

It is... :) you're missing an important set of ( )'s. :)

Answer 9

Remember, you are multiplying that WHOLE RHS by (y-6)

Answer 10

OOPS, LHS

Answer 11

(all was fine with what you did on the RHS)

Answer 12

\[(y-6)(x+4) = -1\]

wouldnt it then look like
\[yx+4y-6x-24 = -1\]

Answer 13

yes, good so far. Now remember your goal: solve for y.

So keep the terms with y's on the LHS. Move anything without a y to the RHS.

Answer 14

Everything ok? (You must be typing a really long reply... lol :)

Answer 15

This equation makes me nervous D:

\[yx+4y -6x -24 = -1\]

\[yx + 4y = 6x +23\]

\[\frac{ yx + 4x }{ x } = \frac{ 6x + 23 }{ x }\]

\[y + \frac{ 4y }{ x } = 29 \]
um i think i went wrong somewhere

Answer 16

lol, sorry, im getting used to this insert equation thing

Answer 17

That's ok, I was just on the edge of my seat! :) lol

OK you were fine here:

\(yx + 4y = 6x +23\) Perfect!

Now.... you are trying to SOLVE FOR y. Dividing by x is not useful here... you have 2 terms on the LHS with a y, so you want to get y "alone", were you will only "see" ONE y.

How to do that? Maybe..... factor out that y, on the LHS?

\(y(x + 4) = 6x +23\)

Do you see the difference, in that vs. dividing by x? Now I only have the variable y, once. I just need to finish solving for it! So what's the next step?

Answer 18

yes, thank you! i totally forgot about factoring :/

Answer 19

Never, ever, forget about factoring. :)

Answer 20

does this equation look simplified to you?\[y = \frac{ 6x + 23 }{ x + 4 }\]

Answer 21

Yes, that's fine! That's the inverse.

Answer 22

you can re-write it in the inverse notation:

\(\Large f^{-1}(x) = \dfrac{ 6x + 23 }{ x + 4 }\)

Answer 23

Now, do you know how to go about graphing it?

You'll want to think about:
- vertical asymptotes (and behavior near them)
- horizontal asymptote
- x & y intercepts

Those should give you a pretty good understanding of the graph. :)

Answer 24

I graphed it into my graphing calculate but it looks like a straight line. i think its not a function though maybe thats why it wont graph properly on my calculator

Answer 25

It is a function. It might be your viewing window.

Answer 26

Adjust your view, you may just be zoomed in too close. :)

Answer 27

I was following along, Debbie and you did a great job working together . this is desmos

Answer 28

ah, i was in standard zoom

Answer 29

thanks @zupster.. and lol, I was just preparing something similar to show her.. :)

see, the cool thing about a function and it's inverse is, if you look at them together, they are reflections of one another over the line y=x.

Answer 30

wow. im so sorry. when i wrote the equation i forgot about the horizontal compression. give me a moment to resolve it

Answer 31

http://kevinmehall.net/p/equationexplorer/#y%20=%20%286%20x+23%29/%28x+4%29 |y%20=%20-1/%28x-6%29-4|y=x|[-31.622776601683793,31.622776601683793,-31.622776601683793,31.622776601683793]

Answer 32

Well, that link didn't work out too well, but you can c&p it into a new tab. :) It's cool. :)

Answer 33

I like it better with this view:

http://kevinmehall.net/p/equationexplorer/#y%20=%20%286%20x+23%29/%28x+4%29 |y%20=%20-1/%28x-6%29-4|y=x|[-10,15,-10,15]

Answer 34

i ended up with \[3y(x+4) = 6x + 23\]

\[\frac{ 3y }{ 3 } = \frac{ 6x=23 }{ x+4 } \]
i got stuck again

Answer 35

would it be multiplied by 3?

Answer 36

agreed, asymptotes should be shown

Answer 37

Just multiply on RHS by (1/3) :)

Answer 38

You divided by 3 on the LHS, now you need to divide by 3 on the RHS, right?

But easier to "execute" that, if you think of it as multiplying by 1/3 (because, after all.... that's exactly what division by 3, IS :)

Answer 39

\[\frac{ 6x+23 }{ 3(x+4) }\]

Answer 40

does that look right?

Answer 41

From what you had above, yes. I'm not sure what your starting function was here...?

Answer 42

f(x) = \[-\frac{ 1 }{ 3x-6 } -4\]

Answer 43

sorry, i forgot about the horizontal compression applied to the parent function so i forgot to add the 3 to the original equation

Answer 44

That's OK, our first one was a good learning exercise, then. :)

Answer 45

yes, thank you so much!

Answer 46

Sure thing, happy to help. :)