Need help finding an inverse function

I have a function that describes a ball being thrown at an initial velocity of 192 ft/s, given by:

**f(t) = 192t - 16t^2**

I need to find an inverse function that yields the time 't' when the ball is at a height 'h' as the object travels upward:

**t = 6 - [sqrt(576 - h) / 4]**

Also, I need to find an inverse function that yields the time 't' when the ball is at height 'h' as the object travels downward:

**t = 6 + [sqrt(576 - h) / 4]**

The problem is that I can't figure out how to find these inverse functions from **f(t) = 192t - 16t^2**. I tried factoring, but didn't have any luck. Also, since this is a second order function, it's not one-to-one, which means that it shouldn't have an inverse, right (since it will fail the horizontal line test)? The two answers that were given above were what was listed, but I can't figure out how to get to the answer. (Worried)

Help would be greatly appreciated! Thanks!

Re: Need help finding an inverse function

When the ball is at height $\displaystyle h$ that means $\displaystyle f(t)=h$ so the equation becomes:

$\displaystyle h=192t-16t^2$

$\displaystyle \Leftrightarrow 16t^2-192t+h=0$

Calculating the discriminant gives:

$\displaystyle D=(-192)^2-4\cdot 16 \cdot h = 36 864 - 64h=64(576-h)$

So the two solutions are ...

Re: Need help finding an inverse function

Siron is referring to the "quadratic formula".

Solutions to $\displaystyle ax^2+ bx+ c= 0$ are given by

$\displaystyle \frac{-b\pm\sqrt{b^2- 4ac}}{2a}$.

Your function, $\displaystyle h= 192t- 16t^2$ is the same as $\displaystyle 16t^2- 192t+ h= 0$ so a= 16, b= -192, c= h.

Re: Need help finding an inverse function

Or you can use the "completing the square" method.

First make 16t^2-192t the subject:

16t^2-192t = -h

Factorise to make it easier:

16(t^2-12t) = -h

Completing the square gives:

16((t-6)^2-36) = -h

You should be able to take it from here...

It's the same as HallofIvy's only different approach...