{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?

검증된 단계별 안내

Step 1: Understand the function given: \( h(t) = 64t - 16t^2 \). This is a quadratic function representing the height of a baseball over time.

Step 2: Recall that a function is one-to-one if it passes the horizontal line test, meaning no horizontal line intersects the graph of the function more than once.

Step 3: Analyze the function \( h(t) = 64t - 16t^2 \). This is a downward-opening parabola because the coefficient of \( t^2 \) is negative.

Step 4: Determine the vertex of the parabola, which is the maximum point, using the formula \( t = -\frac{b}{2a} \) where \( a = -16 \) and \( b = 64 \).

Step 5: Evaluate whether the function is increasing or decreasing on the interval \( 0 \leq t \leq 4 \) by checking the behavior of the function before and after the vertex within this interval.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

{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?