Sure, let's analyze the given piecewise function [tex]\( f(x) \)[/tex] and evaluate it for the specific points to check the given statements.
The piecewise function is:
[tex]\[
f(x) = \left\{
\begin{array}{ll}
-3x, & \text{if } x < 0 \\
4, & \text{if } x = 0 \\
x^2, & \text{if } x > 0
\end{array}
\right.
\][/tex]
Let's evaluate the function at the given points and check the corresponding statements:
1. Evaluate [tex]\( f(1) \)[/tex]:
Since [tex]\( 1 > 0 \)[/tex], we use the piece [tex]\( x^2 \)[/tex]:
[tex]\[
f(1) = 1^2 = 1
\][/tex]
The statement [tex]\( f(1) = -3 \)[/tex] is:
[tex]\[
\text{False}
\][/tex]
2. Evaluate [tex]\( f(3) \)[/tex]:
Since [tex]\( 3 > 0 \)[/tex], we use the piece [tex]\( x^2 \)[/tex]:
[tex]\[
f(3) = 3^2 = 9
\][/tex]
The statement [tex]\( f(3) = 9 \)[/tex] is:
[tex]\[
\text{True}
\][/tex]
3. Evaluate [tex]\( f(2) \)[/tex]:
Since [tex]\( 2 > 0 \)[/tex], we use the piece [tex]\( x^2 \)[/tex]:
[tex]\[
f(2) = 2^2 = 4
\][/tex]
The statement [tex]\( f(2) = 4 \)[/tex] is:
[tex]\[
\text{True}
\][/tex]
4. Evaluate [tex]\( f(4) \)[/tex]:
Since [tex]\( 4 > 0 \)[/tex], we use the piece [tex]\( x^2 \)[/tex]:
[tex]\[
f(4) = 4^2 = 16
\][/tex]
The statement [tex]\( f(4) = 0 \)[/tex] is:
[tex]\[
\text{False}
\][/tex]
Based on these evaluations, the true statements are:
- [tex]\( f(3) = 9 \)[/tex]
- [tex]\( f(2) = 4 \)[/tex]
The false statements are:
- [tex]\( f(1) = -3 \)[/tex]
- [tex]\( f(4) = 0 \)[/tex]
