When the ball is at height that means so the equation becomes:
Calculating the discriminant gives:
So the two solutions are ...
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.
Help would be greatly appreciated! Thanks!
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...