Question

The height in feet of a baseball hit straight up from the ground with an initial velocity of \(64 \mathrm{ft} / \mathrm{s}\) is given by \(h=f(t)=64 t-16 t^{2},\) where \(t\) is measured in seconds after the hit. a. Is this function one-to-one on the interval \(0 \leq t \leq 4 ?\) b. Find the inverse function that gives the time \(t\) at which the ball is at height \(h\) as the ball travels upward. Express your answer in the form \(t=f^{-1}(h)\) c. Find the inverse function that gives the time \(t\) at which the ball is at height \(h\) as the ball travels downward. Express your answer in the form \(t=f^{-1}(h)\) d. At what time is the ball at a height of \(30 \mathrm{ft}\) on the way up? e. At what time is the ball at a height of \(10 \mathrm{ft}\) on the way down?

Short answer

Answer: No, the function is not one-to-one on the given interval.

Step-by-step solution

Determine if the function is increasing or decreasing on the given interval

To determine if the function is one-to-one, we need to check whether the function is strictly increasing or strictly decreasing on the interval \(0 \leq t \leq 4\). To do that, we will find the first derivative of \(h(t)\): $$h'(t) = \frac{d}{dt}(64t - 16t^2) = 64 - 32t.$$

Check for one-to-one on the interval

Now, evaluate the first derivative on the interval \([0, 4]\) to see if its sign is consistent: \(h'(0) = 64 > 0\), the function is increasing at \(t = 0\). \(h'(4) = 64 - 32(4) = -64 < 0\), the function is decreasing at \(t=4\). Since the function is increasing for part of the interval and decreasing for the other part, the function is not one-to-one on the given interval. #b-Finding Inverse Function for Upward Motion#

Solve the quadratic for \(t\)

In order to find the inverse function for traveling upward, we will solve the quadratic for \(t\): $$h = 64t - 16t^2 = 16t(4-t)$$ Since the upward motion corresponds to when the ball first reaches the height of \(h\), we only consider the root of the equation \(h = 16t(4-t)\) when \(t \leq 4\). Rearrange the equation to find t: $$t(4-t) = \frac{h}{16}$$

Quadratic Formula

Apply the quadratic formula with \(a=-1, b=4, c=-\frac{h}{16}\): $$t = \frac{-b\pm\sqrt{b^2-4ac}}{2a} = \frac{-4 \pm \sqrt{16 + h}}{2(-1)}$$

Choose the correct root

We choose \(t = 4 - \sqrt{16 + h}\) since it is the smaller solution and corresponds to the upward motion: $$t = f^{-1}(h) = 4 - \sqrt{16 + h}$$ #c-Finding Inverse Function for Downward Motion#

Choose the other root

We choose the second root of the quadratic from part b: $$t = f^{-1}(h) = 4 + \sqrt{16 + h}$$ #d- Time at a height of 30 ft on the way up#

Use the inverse function for upward

Use the inverse function for upward motion to find the time when \(h = 30\): $$t = f^{-1}(30) = 4 - \sqrt{16+30}$$ $$t \approx 1.69 \;\mathrm{s}$$ #e- Time at a height of 10 ft on the way down#

Use the inverse function for downward

Use the inverse function for downward motion to find the time when \(h = 10\): $$t = f^{-1}(10) = 4 + \sqrt{16+10}$$ $$t \approx 4.81 \;\mathrm{s}$$

Key concepts

One-to-One Function

A function is considered one-to-one if it passes both the horizontal and vertical line tests. This means that for every input, there is a unique output, and no two different inputs map to the same output. In simpler terms, a function is one-to-one if no y-value is repeated in the function's range.
To explore if a given function like \( h = f(t) = 64t - 16t^2 \) is one-to-one over an interval, we need to check whether the function is strictly increasing or strictly decreasing over that interval. This ensures that each y-value corresponds to only one x-value.
For this particular function, by evaluating the first derivative \( h'(t) = 64 - 32t \), we can assess the behavior of the function in the interval \([0, 4]\). Since the derivative changes sign within this range (positive at \( t=0 \) and negative at \( t=4 \)), the function is not strictly increasing or decreasing throughout the interval, indicating that the function is not one-to-one on the entire interval [0, 4].

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). It allows us to find the roots of the quadratic equation by using the formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] This formula provides two solutions, known as roots, which may be real or complex.

• The term \( b^2 - 4ac \) is known as the discriminant and helps determine the nature of the roots.
• If the discriminant is positive, the quadratic has two distinct real roots.
• If zero, it has exactly one real root (a repeated root).
• If negative, the roots are complex.

For instance, when determining the inverse of the quadratic function \( h = 64t - 16t^2 \), the quadratic formula is used to isolate \( t \). By setting \( h = 64t - 16t^2 \), rearranging to match the standard quadratic form, and applying the formula, we can find \( t \) values corresponding to specific heights \( h \).

First Derivative

The first derivative of a function provides us with the slope of the tangent line at any given point. It indicates how the function's output changes as its input changes. Essentially, the derivative tells us whether the function is increasing or decreasing at particular points.
The first derivative is particularly useful in analyzing functions to determine where they are increasing, decreasing, constant, or have any critical points. For the function \( h(t) = 64t - 16t^2 \), the first derivative is \( h'(t) = 64 - 32t \).
Assessing this derivative within the interval \([0, 4]\) helps us understand the motion of the baseball. At \( t=0 \), the derivative is positive, meaning the function is increasing (the ball is moving upward). As \( t \) increases toward 4, the derivative becomes negative, indicating the ball has started its descent. This change confirms the parabolic path typical of objects under constant negative acceleration (like gravity).

Upward and Downward Motion

The concepts of upward and downward motion are pivotal in understanding the parabolic motion of objects thrown into the air, like a baseball in this scenario. Upward motion refers to the ball rising to a point, while downward motion describes its fall back to the ground.
In a quadratic height function such as \( h(t) = 64t - 16t^2 \), the motion can be analyzed by finding points where the function has a zero derivative. These points are typically maximums or minimums in the function's graph.
In this context:

• The upward motion lasts until the ball reaches its peak height. It corresponds to the segment of the function where the derivative is positive, and the ball's height is increasing.
• The downward motion begins after the peak and continues as the derivative turns negative, indicating the height is decreasing.

Using inverse functions derived from solving the quadratic allows us to find specific times for a given height during either phase of motion. For example, using the upward inverse function, time can be calculated for specific heights reached while ascending.