Let's evaluate the piecewise function [tex]\( f(x) \)[/tex] at the specified points to determine which statements are true. The piecewise function is defined as follows:
[tex]\[
f(x)=\left\{\begin{array}{c}
-x+1, \quad x<0 \\
-2, \quad x=0 \\
x^2-1, \quad x>0
\end{array}\right\}
\][/tex]
1. Evaluating [tex]\( f(-2) \)[/tex]:
Since [tex]\( -2 < 0 \)[/tex], we use the first part of the piecewise function [tex]\( f(x) = -x + 1 \)[/tex]:
[tex]\[
f(-2) = -(-2) + 1 = 2 + 1 = 3
\][/tex]
So, [tex]\( f(-2) = 3 \)[/tex]. Therefore, statement A, [tex]\( f(-2) = 0 \)[/tex], is incorrect.
2. Evaluating [tex]\( f(4) \)[/tex]:
Since [tex]\( 4 > 0 \)[/tex], we use the third part of the piecewise function [tex]\( f(x) = x^2 - 1 \)[/tex]:
[tex]\[
f(4) = 4^2 - 1 = 16 - 1 = 15
\][/tex]
So, [tex]\( f(4) = 15 \)[/tex]. Therefore, statement B, [tex]\( f(4) = 7 \)[/tex], is incorrect.
3. Evaluating [tex]\( f(-1) \)[/tex]:
Since [tex]\( -1 < 0 \)[/tex], we use the first part of the piecewise function [tex]\( f(x) = -x + 1 \)[/tex]:
[tex]\[
f(-1) = -(-1) + 1 = 1 + 1 = 2
\][/tex]
So, [tex]\( f(-1) = 2 \)[/tex]. Therefore, statement C, [tex]\( f(-1) = 2 \)[/tex], is correct.
4. Evaluating [tex]\( f(1) \)[/tex]:
Since [tex]\( 1 > 0 \)[/tex], we use the third part of the piecewise function [tex]\( f(x) = x^2 - 1 \)[/tex]:
[tex]\[
f(1) = 1^2 - 1 = 1 - 1 = 0
\][/tex]
So, [tex]\( f(1) = 0 \)[/tex]. Therefore, statement D, [tex]\( f(1) = 0 \)[/tex], is correct.
In summary, the true statements are:
- Statement C: [tex]\( f(-1) = 2 \)[/tex]
- Statement D: [tex]\( f(1) = 0 \)[/tex]
Therefore, the correct statements are C and D.
