Answer:
[tex]h(t)[/tex] is likely a quadratic function.
Based on values in the table, domain of [tex]h(t)[/tex] : [tex]\lbrace 0,\, 1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, 8\rbrace[/tex]; range of [tex]h(t)\![/tex]: [tex]\lbrace 0,\, 35.1,\, 60.1\, 75.2,\, 80.3,\, 75.3,\, 60.2,\, 35.0 \rbrace[/tex].
Step-by-step explanation:
By the power rule, [tex]h(t)[/tex] is a quadratic function if and only if its first derivative, [tex]h^\prime(t)[/tex], is linear.
In other words, [tex]h(t)[/tex] is quadratic if and only if [tex]h^\prime(t)[/tex] is of the form [tex]a\, x + b[/tex] for some constants [tex]a[/tex] and [tex]b[/tex]. Tables of differences of [tex]h(t)\![/tex] could help approximate whether [tex]h^\prime(t)\![/tex] is indeed linear.
Make sure that values of [tex]t[/tex] in the first row of the table are equally spaced. Calculate the change in [tex]h(t)[/tex] over each interval:
- [tex]h(1) - h(0) = 35.1[/tex].
- [tex]h(2) - h(1) = 25.0[/tex].
- [tex]h(3) - h(2) = 15.1[/tex].
- [tex]h(4) - h(3) = 5.1[/tex].
- [tex]h(5) - h(4) = -5.0[/tex].
- [tex]h(6) - h(5) = -15.1[/tex].
- [tex]h(7) - h(6) = -25.2[/tex].
- [tex]h(8) - h(7) = -35.0[/tex].
Consecutive changes to the value of [tex]h(t)[/tex] appears to resemble a line with slope [tex](-10)[/tex] within a margin of [tex]0.2[/tex]. Hence, it is likely that [tex]h(t)\![/tex] is indeed a quadratic function of [tex]t[/tex].
The domain of a function is the set of input values that it accepts. For the [tex]h(t)[/tex] of this question, the domain of [tex]h(t)\![/tex] is the set of values that [tex]t[/tex] could take. These are listed in the first row of this table.
On the other hand, the range of a function is the set of values that it outputs. For the [tex]h(t)[/tex] of this question, these are the values in the second row of the table.
Since both the domain and range of a function are sets, their members are supposed to be unique. For example, the number "[tex]0[/tex]" appears twice in the second row of this table: one for [tex]t = 0[/tex] and the other for [tex]t = 8[/tex]. However, since the range of [tex]h(t)[/tex] is a set, it should include the number [tex]0\![/tex] only once.
