{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t²  where t is measured in seconds after the hit.


a. Is this function one-to-one on the interval 0 ≤ t ≤ 4?

Key Concepts

One-to-One Function

A function is considered one-to-one if it assigns distinct outputs to distinct inputs, meaning that no two different inputs produce the same output. To determine if a function is one-to-one, we can use the horizontal line test: if any horizontal line intersects the graph of the function more than once, the function is not one-to-one. In the context of the given height function, analyzing its behavior over the specified interval is crucial.

Quadratic Functions

The function given, h(t) = 64t - 16t², is a quadratic function, which typically has a parabolic shape. Quadratic functions can open upwards or downwards depending on the sign of the leading coefficient. In this case, since the coefficient of t² is negative, the parabola opens downwards, indicating that the function will reach a maximum height before decreasing, which is important for understanding its behavior over the interval.

Critical Points and Intervals

Critical points of a function occur where its derivative is zero or undefined, indicating potential local maxima or minima. For the height function, finding the derivative and setting it to zero will help identify critical points within the interval [0, 4]. Analyzing these points will reveal whether the function is increasing or decreasing, which is essential for determining if it is one-to-one on the specified interval.