Question

Okay last type of question like this then I have equalities which *fingerscrossed* should be easier and I won't have to pester openstudy anymore for the day xD

Answer 1

Based on the last question like this, I need to solve for x first, is that correct?

Answer 2

@DebbieG Thanks c: If you can't its no problem

Answer 3

Well, it's worded kind of weirdly to be honest. But I think it is asking you to find \(f^{-1}(-1)\), using the function f(x).

Now, here's the thing about functions and inverses: remember, the domain and range are flipped, right? So \(f^{-1}(-1)\) takes as its INPUT (domain element), -1, and gives as its OUTPUT (range element), \(f^{-1}(-1)\).

But that OUTPUT of the inverse function is the same as the INPUT (x value) of the ORIGINAL function, that gives as the OUTPUT, y=-1, right?

So you don't EVEN NEED to actually FIND the inverse function \(f^{-1}(x)\). All you need, to know what \(f^{-1}(-1)\) is (what output you get), is to find the INPUT of the ORIGINAL function f(x), that gives you the output of -1.

That is, SOLVE f(x)=-1, find that x-value, and THAT is the value of \(f^{-1}(-1)\).

Answer 4

Okay, I understand that, do I plug in any number for x?

Answer 5

well not any number, but make educated guesses?

Answer 6

I'm not sure I follow you. You have:

\[\Large f(x)=\dfrac{ -2 }{ 4x-3 }\]

You want \(f^{-1}(-1)\), which is simply the value in the DOMAIN of f that gives f(x)=-1.

E.g.: If \(f^{-1}(-1)=y\) then \(f(y)=-1\)

So find the X that maps f(x) to -1, and you will have \(f^{-1}(-1)\)

Eg: solve:

\[\Large -1=\dfrac{ -2 }{ 4x-3 }\]

for x.... whatever that solution is, that's your value of \(f^{-1}(-1)\).

Answer 7

No, no guesses. Solve the equation! lol :)

Answer 8

ohh I forgot about the equation whoops XD

Answer 9

Mind you, the other option is to FIND the INVERSE, then just evaluate THAT at x=-1. but I think the whole point of this problem is to get you to understand the connection between the domains/ranges of the inverse and the original function.

Answer 10

It would actually be a good mental exercise, to find it BOTH ways. And of course, you should get the same answer! :)

Answer 11

5/4 = x
Is that right? I'm double checking my work

Answer 12

Yes, looks right to me! :)

Answer 13

Awesome:DD Is that my answer?

Answer 14

Well, that IS \(f^{-1}(-1)\)... now, whether it really wants you to FIND the INVERSE FUNCTION is not entirely clear. It isn't necessary to find \(f^{-1}(-1)\) (since, as discussed, the value of \(f^{-1}(-1)\) is also the x that you find when you solve \(f(x)=-1\)). But as the question is stated, I would think that yes, that's all you really need for the answer.

Answer 15

Okey doke! Are these the right steps?
f(x) = -2/4x - 3
-1 = -2/4x - 3
*-2 *-2
2 = 4x - 3
+3 +3
5 = 4x
/4 /4
6/4 = x

Answer 16

that's supposed to be 5/4*

Answer 17

I don't understand what you did in this step:
-1 = -2/4x - 3
*-2 *-2 <------
2 = 4x - 3

You appear to be SAYING that you multiplied by -2?? But that would NOT give you the result in the next step; multiplying both sides by -2 would give you:
2 = 4/(4x - 3 )

What you REALLY did is multiplied both sides by (4x-3), AND multiplied both sides by (-1). THAT gives you
2 = 4x - 3

Answer 18

After that, all of your steps are correct. :)

Answer 19

f(x) = -2/4x - 3
-1 = -2/4x - 3
*(4x - 3) *(4x - 3)
4x + 3 = -2
-3 -3
4x = -5
*-1 *-1
-4x = 5
/-4 /-4
x = -5/4
Is that it?

Answer 20

Notice your sign error? That's because when you multiplied by (4x - 3) , you "lost" the -1 on the LHS, but you did not change the sign on the RHS. In other words, on the LHS you actually multiplied by (-1)(4x - 3) , which took care of the -1 on that side. So on the RHS, you also want to multiply by (-1)(4x - 3), to cancel in the den'r and make the num'r =+2.

You could also do this in 2 separate steps if you prefer, but what you CANT do is multiply by something different on the RHS than on the LHS.

Answer 21

what does LHS and RHS stand for?

Answer 22

I don't get the (-1)(4x - 3) part yet

Answer 23

left hand side, right hand side

Answer 24

ok, wait... you do have an error in that step, but not the one I thought you did, lol:

-1 = -2/4x - 3
*(4x - 3) *(4x - 3)
4x + 3 = -2

-1(4x-3)=-4x+3 NOT 4x+3

But what you really want to do is this:
-1 = -2/(4x - 3) multiply both sides by 4x - 3
*(4x - 3) *(4x - 3)
-1(4x - 3) = -2 <---gives you this, NOW multiply both sides by -1
4x - 3 = 2 now continue from here....

Answer 25

f(x) = -2/4x - 3
-1 = -2/4x - 3
*(4x - 3) *(4x - 3)
-1(4x + 3) = -2
-4x - 3 = -2
*-1 *-1
4x - 3 = 2
+3 +3
4x = 5
/4 /4
x = 5/4

How about that? I think I have it