Sure, let's examine each statement with respect to the piecewise function given:
[tex]\[ f(x) = \begin{cases}
2x & \text{if } x < 1 \\
5 & \text{if } x = 1 \\
x^2 & \text{if } x > 1
\end{cases} \][/tex]
### Checking Statement A: [tex]\( f(-2) = 4 \)[/tex]
For [tex]\( x = -2 \)[/tex]:
- Since [tex]\( -2 < 1 \)[/tex], we use the first piece of the function.
- Calculate [tex]\( f(-2) \)[/tex]:
[tex]\[ f(-2) = 2 \times (-2) = -4 \][/tex]
This is not equal to 4. So, statement A is False.
### Checking Statement B: [tex]\( f(5) = 1 \)[/tex]
For [tex]\( x = 5 \)[/tex]:
- Since [tex]\( 5 > 1 \)[/tex], we use the third piece of the function.
- Calculate [tex]\( f(5) \)[/tex]:
[tex]\[ f(5) = 5^2 = 25 \][/tex]
This is not equal to 1. So, statement B is False.
### Checking Statement C: [tex]\( f(1) = 5 \)[/tex]
For [tex]\( x = 1 \)[/tex]:
- Since [tex]\( x = 1 \)[/tex], we use the second piece of the function.
- Calculate [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 5 \][/tex]
This is indeed equal to 5. So, statement C is True.
### Checking Statement D: [tex]\( f(2) = 4 \)[/tex]
For [tex]\( x = 2 \)[/tex]:
- Since [tex]\( 2 > 1 \)[/tex], we use the third piece of the function.
- Calculate [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = 2^2 = 4 \][/tex]
This is indeed equal to 4. So, statement D is True.
### Summary of Statements:
- Statement A: False
- Statement B: False
- Statement C: True
- Statement D: True
Thus, statements C and D are true, while statements A and B are false.
