Sure, let's evaluate the function [tex]\( f(x) \)[/tex] for specific values of [tex]\( x \)[/tex]. We have three cases depending on the value of [tex]\( x \)[/tex].
1. For [tex]\( -3 \leq x < 0 \)[/tex], we have [tex]\( f(x) = -x^2 + 4 \)[/tex].
2. For [tex]\( 0 \leq x < 3 \)[/tex], we have [tex]\( f(x) = 4 \)[/tex].
3. For [tex]\( x \geq 3 \)[/tex], we have [tex]\( f(x) = -x + 5 \)[/tex].
Now, let's evaluate [tex]\( f(x) \)[/tex] at specific values of [tex]\( x \)[/tex]:
1. Evaluate at [tex]\( x = -3 \)[/tex]:
- Since [tex]\( -3 \leq -3 < 0 \)[/tex], we use [tex]\( f(x) = -x^2 + 4 \)[/tex].
[tex]\[
f(-3) = -(-3)^2 + 4 = -9 + 4 = -5
\][/tex]
2. Evaluate at [tex]\( x = -1 \)[/tex]:
- Since [tex]\( -3 \leq -1 < 0 \)[/tex], we use [tex]\( f(x) = -x^2 + 4 \)[/tex].
[tex]\[
f(-1) = -(-1)^2 + 4 = -1 + 4 = 3
\][/tex]
3. Evaluate at [tex]\( x = 0 \)[/tex]:
- Since [tex]\( 0 \leq 0 < 3 \)[/tex], we use [tex]\( f(x) = 4 \)[/tex].
[tex]\[
f(0) = 4
\][/tex]
4. Evaluate at [tex]\( x = 2 \)[/tex]:
- Since [tex]\( 0 \leq 2 < 3 \)[/tex], we use [tex]\( f(x) = 4 \)[/tex].
[tex]\[
f(2) = 4
\][/tex]
5. Evaluate at [tex]\( x = 3 \)[/tex]:
- Since [tex]\( x \geq 3 \)[/tex], we use [tex]\( f(x) = -x + 5 \)[/tex].
[tex]\[
f(3) = -3 + 5 = 2
\][/tex]
6. Evaluate at [tex]\( x = 4 \)[/tex]:
- Since [tex]\( x \geq 3 \)[/tex], we use [tex]\( f(x) = -x + 5 \)[/tex].
[tex]\[
f(4) = -4 + 5 = 1
\][/tex]
So, the evaluations result in the following:
- [tex]\( f(-3) = -5 \)[/tex]
- [tex]\( f(-1) = 3 \)[/tex]
- [tex]\( f(0) = 4 \)[/tex]
- [tex]\( f(2) = 4 \)[/tex]
- [tex]\( f(3) = 2 \)[/tex]
- [tex]\( f(4) = 1 \)[/tex]
Thus, the calculated outputs for the function [tex]\( f(x) \)[/tex] at the specified values are:
[tex]\[ [-5, 3, 4, 4, 2, 1] \][/tex]
