Question

an object was launched off the top of a building. the function $f(x) = -16x^2 + 48x + 64$ represents the height of the object above the ground, in feet, $x$ seconds after being launched. find and interpret the given function values and determine an appropriate domain for the function.
answer attempt 1 out of 2
$f(-1) = \square$, meaning that seconds after the object was launched, the object was feet above the ground.
this interpretation in the context of the problem.
$f(0.5) = \square$, meaning that seconds after the object was launched, the object was feet above the ground.
this interpretation in the context of the problem.
$f(5) = \square$, meaning that seconds after the object was launched, the object was feet above the ground.
this interpretation in the context of the problem.
based on the observations above, it is clear that an appropriate domain for the function is

Explanation:

Step1: Calculate $f(-1)$

Substitute $x=-1$ into $f(x)$:
$$f(-1) = -16(-1)^2 + 48(-1) + 64$$
$$= -16(1) - 48 + 64$$
$$= -16 - 48 + 64 = 0$$

Step2: Calculate $f(0.5)$

Substitute $x=0.5$ into $f(x)$:
$$f(0.5) = -16(0.5)^2 + 48(0.5) + 64$$
$$= -16(0.25) + 24 + 64$$
$$= -4 + 24 + 64 = 84$$

Step3: Calculate $f(5)$

Substitute $x=5$ into $f(x)$:
$$f(5) = -16(5)^2 + 48(5) + 64$$
$$= -16(25) + 240 + 64$$
$$= -400 + 240 + 64 = -96$$

Step4: Find valid domain

Find when $f(x)=0$ (object hits ground):
$$-16x^2 + 48x + 64 = 0$$
Divide by $-16$:
$$x^2 - 3x - 4 = 0$$
Factor:
$$(x-4)(x+1)=0$$
Solutions: $x=4$ or $x=-1$. Since time can't be negative, the object is in the air from $x=0$ (launch) to $x=4$ (hits ground).

Answer:

$f(-1) = 0$, meaning that -1 seconds after the object was launched, the object was 0 feet above the ground. This interpretation is not meaningful in the context of the problem.
$f(0.5) = 84$, meaning that 0.5 seconds after the object was launched, the object was 84 feet above the ground. This interpretation is meaningful in the context of the problem.
$f(5) = -96$, meaning that 5 seconds after the object was launched, the object was -96 feet above the ground. This interpretation is not meaningful in the context of the problem.
Based on the observations above, it is clear that an appropriate domain for the function is non-negative real numbers from 0 to 4 (inclusive), or $[0,4]$