kaylaprincess one year ago help with simple subject with inverse functions ~ I don't get f^-1(x) ... What does that mean? And how do i continue to solve?

1. kaylaprincess

So I looked up to understand more : take the opposite of the function. 1. Change f(x) to y 2. switch x and y 3. solve for y but what about the -1 ??

2. kaylaprincess

here's an example : Let f(x) = x2 - 16. Find f-1(x)

3. kaylaprincess

wait i think i see what im freaking out about wow lol

4. Empty

Yeah you did a good job of tracking down the steps of _how_ to do this, so I'll explain this so you understand what the heck this does! All the inverse function is, $$f^{-1}(x)$$ is the function that you can plug into the original function to get x. This sounds weird, but here's an example: $$f(x)=x^2$$ $$f^{-1}(x)=\sqrt{x}$$ So now check it out, we can plug them into each other either way: $$f(f^{-1}(x))=(\sqrt{x})^2=x$$ or $$f^{-1}(f(x)) =\sqrt{x^2}= x$$ Don't let the negative sign disturb you, it's just a fancy bit of notation to tell you it's special. It is only to remind you it has this relationship to $$f(x)$$ and doesn't actually mean anything more.

5. Empty

Just to be complete, I'll show how we could have solved this with that method to make it all connect up: $$f(x)=x^2$$ Let's turn $$f(x)$$ into $$x$$ and $$x$$ into $$y$$ : $$x=y^2$$ Solve for $$y$$ $$\sqrt{x}=y$$ Oh hey, this $$y$$ is really our $$f^{-1}(x)$$ we were talking about earlier, we're literally just undoing the function from the standpoint of x, so that's why we call it the inverse.

6. kaylaprincess

so for any final answer in an inverse function question, write it out with the -1? |dw:1437687398734:dw|

7. kaylaprincess

thank you for helping me so much by the way..!!

8. Empty

Yeah for my example it will be but for your example it will be slightly different! $$f(x)=x^2-16$$ Go ahead and try this out by replacing f(x) with x and x with y, you'll get something a little different.

9. kaylaprincess

$16x = y ^{2 }$

10. kaylaprincess

?

11. kaylaprincess

then take the square root ?

12. Empty

You're on the right track, so to get from here to here you have to add, like this: $$x=y^2-16$$ Now we can add 16 to both sides, since that will keep both sides equal to each other, which is what we have to do in order to keep that = sign correctly. $$x+16=y^2 -16+16$$ So now the point of that is because we both know that $$16-16=0$$ on the right side there! $$x+16=y^2$$ Now you can do what you said earlier and take the square root of both sides! :D