Question

Determine whether the given description is of a one-to-one function: f(t) is the height of a football t seconds after kickoff.

The answer is no but idk why.

Answer 1

Since the football goes up and then down, it reaches some height twice. Once while it's moving upwards, and once while it's moving downward. Thus, since it has the same y-value for two x-values, it isn't one-to-one.

Answer 2

But f(t) is at t seconds after kickoff. Based on your answer (I'm thinking of a concave down parabola), f(t) has reached it's maximum point and is returning to the height of however high a football is. Let's assume f(t) is 4-(t-2)^2. Are you saying that t hits both points just above the x-axis (the x-axis being the ground) because the question is asking about kickoff.

Answer 3

In your example, take when t=1, and when t=3. In both cases, f(t)=3. Since both values for t give the same value for f(t), it isn't one-to-one.

Answer 4

So basically, the value of t is indefinite and since there was no restriction, t can be both 1 and 3. Is that what you're saying?

Answer 5

Basically. In formal math terms, since \(f(t_1) = f(t_2)\) does not imply \(t_1 = t_2\) it isn't one-to-one.

Answer 6

Okay, thanks. I actually chose that example since t=0 as the starting point is more realistic. After all, time can't be negative. Other than that, thanks for the explanation.

Answer 7

You're welcome.