{Use of Tech} Height and time The height in feet of a baseball...

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?

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)

Accepted Answer

Welcome back everyone. In this problem, a share market opens at 9 AM and closes at 3 PM. The price of a share for a particle share on a given date is p of x equals 12 x minus 2 x squared, where x is the number of hours from 9 AM. We want to find the inverse of p of x when the value of p of x is increasing. For our answer choices, a says that the inverse of p of x equals 3 plus a half of the square root of 36 minus 2 x. B is 3 minus a half of the square root of 36 minus 2 x. C is 3 minus a half of the square root of 36 plus 2 x. And the d is the is 3 plus a half of the square root of 36 plus 2 x. Now, what do we really want here? We want to find the inverse of p of x. We already know what p of x is. So p of x takes our value of x. Right? And then it gives us a function p of x. It takes the number of hours x and gives us the price for the particle share. But now we want to do the inverse of that. In other words, we want to take the price of a particle share and use that to give us the number of hours for the function. So that's what it means when we're finding the inverse. We're turning the input into the new output and vice versa. So for our function p of x equals 12x minus 2x squared, Let's set Y to P of X. Right. Which now we can write it as an equation. Y equals 12 X minus 2 X squared. And let's also interchange our x's for y's. So now our function becomes x equals 12 y minus 2 y squared. What we're trying to do here is now solve for y. In other words, we're making our input the new output. So let's go ahead and do that. Notice that this is a quadratic equation. So I can rewrite it setting one side of the equation to 0 to put it in its general form. So I can do that by adding 2 y squared and subtracting 12 y from the left hand side of the equation. And writing back x, that means 2 y squared minus 12 y plus x equals 0. Know that it's in its general form. So given a, the coefficient of the squared term equals 2, b, the coefficient of the linear term equals negative 12, and c, the constant equals x. Then by using the quadratic formula, our unknown value y equals negative b, so that's negative negative twelve plus or minus the square root of b squared. That's negative 12 squared minus 4 a, which is 2 c, which is x. And all of that is being divided by 2 multiplied by a, which is 2. So now, we can go ahead and solve for y. Now, negative negative 12 is positive 12. So positive 12 plus or minus negative 12 squared, which is 144, minus 4 multiplied by 2 multiplied by x is negative 8 x. And all of that is being divided by 2 multiplied by 2, which is 4. Now, under our square root, both terms share a common factor of 4. So we can factor 4 out to get the square root of 4 multiplied by 36 minus 2 x. Now, we can use what we know about oh, all of that is being divided by 4. My bad. We can use what we know about the product rule for radicals to help us here because recall that the square root of a product a b equals the square root of a multiplied by the square root of b. And that can help us here because that means we can rewrite our terms under the square root as the square root of 4 multiplied by the square root of 36 minus 2 x and all of that is being divided by 4. Now that helps us because we can solve the square root of 4. 4 is a square number. So we can rewrite that as 12 plus or minus the square root of 4 which is 2 multiplied by 36 minus 2 x all divided by 4. Now let me come over to the left hand side here to make some space. I wish that line would have been straighter, but you know how it is. So now we can go ahead and simplify because each of these terms have 2 as a common factor. Or we can go ahead and divide through by 4. So Y is now going to be equal to 12 divided by 4, which is 3 plus or minus 2 divided by 2 multiplied by the skirt of 36 minus 2 x divided by 4. And 2 goes into 4 two times. So 2 divided by 4 is a half and that is being multiplied by 36 minus 2 x or the square root rather of 36 minus 2 x. Now, remember, what do we know about our function? We want the value for the inverse when p of x is increasing. And the only way p of x would be increasing here is if both values are being added. So we can even rewrite that to say that y equals 3 plus a half of the square root of 36 minus 2 x because if we were finding the difference, then the value would be decreasing. Now that we have solved for y, this is now the inverse of p of x. As in this output or what was once our input is now our new output. So that tells us that we can rewrite it as the inverse of p of x equals 3 plus a half of the square root of 36 minus 2 x. That would be our final answer and when we look in our answer choices, that would have been answer choice a. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Quadratic Functions

Inverse Functions

Projectile Motion

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