To solve for the values of the function [tex]\( f(x) = x^2 - 3 \)[/tex] over the domain [tex]\( D = \{-3, -2, -1, 0, 1, 2, 3\} \)[/tex], we will evaluate the function at each value in the domain step-by-step.
1. Evaluate [tex]\( f(-3) \)[/tex]:
[tex]\[
f(-3) = (-3)^2 - 3 = 9 - 3 = 6
\][/tex]
2. Evaluate [tex]\( f(-2) \)[/tex]:
[tex]\[
f(-2) = (-2)^2 - 3 = 4 - 3 = 1
\][/tex]
3. Evaluate [tex]\( f(-1) \)[/tex]:
[tex]\[
f(-1) = (-1)^2 - 3 = 1 - 3 = -2
\][/tex]
4. Evaluate [tex]\( f(0) \)[/tex]:
[tex]\[
f(0) = 0^2 - 3 = 0 - 3 = -3
\][/tex]
5. Evaluate [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 1^2 - 3 = 1 - 3 = -2
\][/tex]
6. Evaluate [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = 2^2 - 3 = 4 - 3 = 1
\][/tex]
7. Evaluate [tex]\( f(3) \)[/tex]:
[tex]\[
f(3) = 3^2 - 3 = 9 - 3 = 6
\][/tex]
Now, let’s compile all the results:
[tex]\[
\begin{align*}
f(-3) & = 6, \\
f(-2) & = 1, \\
f(-1) & = -2, \\
f(0) & = -3, \\
f(1) & = -2, \\
f(2) & = 1, \\
f(3) & = 6.
\end{align*}
\][/tex]
So, the list of the values of [tex]\( f(x) \)[/tex] for each [tex]\( x \)[/tex] in the domain [tex]\( D \)[/tex] is:
[tex]\[
[6, 1, -2, -3, -2, 1, 6]
\][/tex]
