Certainly! Let's evaluate the function [tex]\( h(t) = -t^2 - 7 \)[/tex] at the given values step by step.
### a. Evaluate [tex]\( h(3) \)[/tex]:
1. Substitute [tex]\( t = 3 \)[/tex] into the function [tex]\( h(t) \)[/tex]:
[tex]\[
h(3) = -(3)^2 - 7
\][/tex]
2. Calculate [tex]\( 3^2 \)[/tex]:
[tex]\[
3^2 = 9
\][/tex]
3. Multiply by [tex]\(-1\)[/tex]:
[tex]\[
-(3^2) = -9
\][/tex]
4. Subtract 7 from [tex]\(-9\)[/tex]:
[tex]\[
-9 - 7 = -16
\][/tex]
Thus,
[tex]\[
h(3) = -16
\][/tex]
### b. Evaluate [tex]\( h(-4) \)[/tex]:
1. Substitute [tex]\( t = -4 \)[/tex] into the function [tex]\( h(t) \)[/tex]:
[tex]\[
h(-4) = -(-4)^2 - 7
\][/tex]
2. Calculate [tex]\((-4)^2 \)[/tex]:
[tex]\[
(-4)^2 = 16
\][/tex]
3. Multiply by [tex]\(-1\)[/tex]:
[tex]\[
-(16) = -16
\][/tex]
4. Subtract 7 from [tex]\(-16\)[/tex]:
[tex]\[
-16 - 7 = -23
\][/tex]
Thus,
[tex]\[
h(-4) = -23
\][/tex]
### c. Evaluate [tex]\( h(0) \)[/tex]:
1. Substitute [tex]\( t = 0 \)[/tex] into the function [tex]\( h(t) \)[/tex]:
[tex]\[
h(0) = -(0)^2 - 7
\][/tex]
2. Calculate [tex]\( 0^2 \)[/tex]:
[tex]\[
0^2 = 0
\][/tex]
3. Multiply by [tex]\(-1\)[/tex]:
[tex]\[
-(0) = 0
\][/tex]
4. Subtract 7 from [tex]\( 0 \)[/tex]:
[tex]\[
0 - 7 = -7
\][/tex]
Thus,
[tex]\[
h(0) = -7
\][/tex]
In summary:
a. [tex]\( h(3) = -16 \)[/tex]
b. [tex]\( h(-4) = -23 \)[/tex]
c. [tex]\( h(0) = -7 \)[/tex]
