Juliana has created the function f(x) = 3x + 2/ 4 to represent the cost of texting on her current plan, where x represents the number of texts. Juliana discovers that using the inverse function to solve for x = 24, she can predict how many texts she can use for $24. Explain to Juliana how to accomplish this, using complete sentences.

Question

anonymous · Answer

So far, I replaced the f(x) with y and then I switched y and x 
$$y = \frac{ 3x+2 }{ 4 }$$

anonymous · Answer

then I switched the y and x and then multiplied both sides by 4 
$$(4)x = \frac{ 3y +2 }{ 4 } (4)$$

anonymous · Answer

then, after canceling out the fraction, I subtracted 2 on both sides --> 4x - 2 = 3y. Now i'm not sure if I should replace the x value with 24 or if i'm doing this all wrong?

anonymous · Answer

@Hero

anonymous · Answer

@Nurali

hero · Answer

Divide both sides by 3 now. Then change $y$ to $f^{-1}(x)$

anonymous · Answer

Where would I divide 3? so far I got 4x - 2 = 3y

hero · Answer

$$\frac{4x - 2}{3} = y$$

anonymous · Answer

Wait, since it started out as f(x) = 3x +2 over 4, to make it an inverse you change the equation? how do you know the equation is like that though?

hero · Answer

How do you know it is the inverse? Well, there is one way to know for sure. Compose both functions. If the result you get is x, then you have the correct inverse.

anonymous · Answer

I still don't understand, what do I do now that I have 4x - 2/ 3 =y? do I multiply by 3 on both sides?

hero · Answer

You have isolated y, which is the goal after swapping x and y. Now you replace y with $f^{-1}(x)$

hero · Answer

$f^{-1}(x)$ stands for the inverse of $f(x)$

anonymous · Answer

So you are telling me that the answer is 4x - 2/ 3 = y?? is there no more?

anonymous · Answer

what about x = 24

hero · Answer

I didn't say you were done with the word problem. I said, you are done finding the inverse once you replace y with $f^{-1}(x)$

hero · Answer

You can continue solving the word problem now that you have the inverse.

anonymous · Answer

So would I now fill in x with 24?

anonymous · Answer

If I filled in x with 24 my answer would be 31.3333 and I don't think that 31.3333 is correct right?

hero · Answer

It is correct, but of course you round to 31

anonymous · Answer

Really? so it would be 31m= F-1(x)? Would I say that her answer is 31?

anonymous · Answer

*31= f-1(x)

hero · Answer

You would say, the number of text she can make for $24 is 31 texts.

anonymous · Answer

Ahh ok, thank you very very much :)