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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 65b
Chapter 1, Problem 65b

{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.


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 = ƒ⁻¹ (h)

Verified step by step guidance
1
Step 1: Start with the given function for height: \( h = f(t) = 64t - 16t^2 \).
Step 2: To find the inverse function, solve for \( t \) in terms of \( h \). Begin by setting \( h = 64t - 16t^2 \).
Step 3: Rearrange the equation to form a standard quadratic equation: \( 16t^2 - 64t + h = 0 \).
Step 4: Use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve for \( t \), where \( a = 16 \), \( b = -64 \), and \( c = h \).
Step 5: Since we are interested in the time when the ball is traveling upward, choose the solution with the positive square root: \( t = \frac{64 + \sqrt{64^2 - 4 \cdot 16 \cdot h}}{2 \cdot 16} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quadratic Functions

The height function h(t) = 64t - 16t² is a quadratic function, which is characterized by its parabolic shape. Quadratic functions can be expressed in the standard form ax² + bx + c, where a, b, and c are constants. In this case, the coefficient of t² is negative, indicating that the parabola opens downward, which is typical for projectile motion.
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Inverse Functions

An inverse function essentially reverses the effect of the original function. For a function f(t), the inverse f⁻¹(h) allows us to find the input t for a given output h. To find the inverse of a quadratic function, we typically solve for t in terms of h, which may involve rearranging the equation and applying the quadratic formula.
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Projectile Motion

Projectile motion describes the motion of an object thrown into the air, influenced only by gravity after its initial launch. The height of the object over time can be modeled by a quadratic equation, where the initial velocity and gravitational acceleration determine the trajectory. Understanding this concept is crucial for interpreting the height function and its inverse in the context of the baseball's flight.
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Related Practice
Textbook Question

{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.


e. At what time is the ball at a height of 10 ft on the way down? 

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Textbook Question

{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?

Textbook Question

Use shifts and scalings to graph the given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed.


h(x)=−4x2−4x+12h(x)=-4x^{^2}-4x+12

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Textbook Question

{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.


d. At what time is the ball at a height of 30 ft on the way up?

Textbook Question

{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.


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 = ƒ⁻¹ (h)

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Textbook Question

Simplify the difference quotient ƒ(x+h)-ƒ(x)/h

ƒ(x) = 4x-3