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

- anonymous

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

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- schrodinger

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- DebbieG

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

- anonymous

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.

- anonymous

inverse*

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## More answers

- DebbieG

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 \(\large y=f^{-1} (x)\)

- anonymous

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

- DebbieG

Yes, that sounds like a good start. :)

- anonymous

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

- DebbieG

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

- DebbieG

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

- DebbieG

OOPS, LHS

- DebbieG

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

- anonymous

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

- DebbieG

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.

- DebbieG

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

- anonymous

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

- anonymous

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

- DebbieG

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?

- anonymous

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

- DebbieG

Never, ever, forget about factoring. :)

- anonymous

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

- DebbieG

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

- DebbieG

you can re-write it in the inverse notation:
\(\Large f^{-1}(x) = \dfrac{ 6x + 23 }{ x + 4 }\)

- DebbieG

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. :)

- anonymous

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

- DebbieG

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

- DebbieG

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

- zpupster

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

##### 1 Attachment

- anonymous

ah, i was in standard zoom

- DebbieG

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.

- anonymous

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

- DebbieG

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]

- DebbieG

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

- DebbieG

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]

- anonymous

i ended up with \[3y(x+4) = 6x + 23\]
\[\frac{ 3y }{ 3 } = \frac{ 6x=23 }{ x+4 } \]
i got stuck again

- anonymous

would it be multiplied by 3?

- zpupster

agreed, asymptotes should be shown

- DebbieG

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

- DebbieG

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 :)

- anonymous

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

- anonymous

does that look right?

- DebbieG

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

- anonymous

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

- anonymous

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

- DebbieG

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

- anonymous

yes, thank you so much!

- DebbieG

Sure thing, happy to help. :)

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