If F(x)=2x-1/3, and F^-1 is the inverse of F, what is the value of F^-1 (3) ?

Question

If F(x)=2x-1/3, and F^-1 is the inverse of F, what is the value of F^-1 (3)  ?

whpalmer4 · Answer

Do you know how to find the inverse of F?

amtran_bus · Answer

That would be a good place to start.

amtran_bus · Answer

I need help.

whpalmer4 · Answer

Okay.  well, let's do a different problem as an example.  $F(x) = 3x-2$

To find the inverse function, $F^{-1}(x)$, we solve for $x$, then swap the variables.

$$y = F(x) = 3x-2$$$$y=3x-2$$$$y+2=3x$$$$x=\frac{y+2}{3}$$Now swap the variables:$$F^{-1}(x) = \frac{x+2}{3}$$

amtran_bus · Answer

Got that.

amtran_bus · Answer

on my problem would the inverse by y=6x-3?

whpalmer4 · Answer

One interesting thing about inverse functions is that if you graph a function and its inverse on the same graph, the inverse is just the function reflected across the line y = x.  Here's the graph of the example I just did:

whpalmer4 · Answer

So I'm not positive about your original function: is it
$$F(x) = \frac{2x-1}{3}$$

amtran_bus · Answer

Sorry, yes it is.

whpalmer4 · Answer

By my figuring, the inverse would then be $y = \frac{1}{2}(1+3x)$

amtran_bus · Answer

Do you mind showing me how?

whpalmer4 · Answer

$$y = \frac{2x-1}{3}$$Solve for $x$:
$$3y = 2x-1$$$$3y+1=2x$$$$\frac{3y+1}{2} = x$$Switch variables:$$F^{-1}(x) = \frac{3x+1}{2} = \frac{1}{2}(3x+1)$$

amtran_bus · Answer

Got ya

whpalmer4 · Answer

Graphed, the functions look like this:

whpalmer4 · Answer

There's that reflection again...if you don't get that result, there's a mistake somewhere.

So now having found the inverse function, what remains to be done is to evaluate it at $x = 3$.  Do that and tell me the answer you get...

amtran_bus · Answer

When x=3, I got 5.

whpalmer4 · Answer

The olive colored line is the $F^{-1}(x)$ graph in my second attachment.  Does the graph match up with your result?

amtran_bus · Answer

Yes.

amtran_bus · Answer

One more thing.  How did you get 1/2 (3x+1) from 3x+1/2?

amtran_bus · Answer

Who, nevermind.  See it!

whpalmer4 · Answer

Excellent!

Okay, I'm going to remind you that (3x+1)/2 does not equal 3x+1/2....

whpalmer4 · Answer

it's a pain putting in the extra parentheses, I agree, but probably 1/3 of the problems I help with on OS start out with me (or someone else) asking for clarification of the equation(s) for just this reason :-)

amtran_bus · Answer

Thanks. God bless.

whpalmer4 · Answer

got any more like to do?

amtran_bus · Answer

No, I am reviewing for a accuplacer test to clep college algebra. I know this stuff, its just been awhile!

whpalmer4 · Answer

Great!  Good luck with the test...

amtran_bus · Answer

Thank you.