Given the piecewise function shown below, select all of the statements that are true.

[tex]\[
f(x)=
\begin{cases}
2x & \text{if } x \ \textless \ 1 \\
5 & \text{if } x = 1 \\
x^2 & \text{if } x \ \textgreater \ 1
\end{cases}
\][/tex]

A. [tex]\( f(-2) = 4 \)[/tex]

B. [tex]\( f(5) = 1 \)[/tex]

C. [tex]\( f(1) = 5 \)[/tex]

D. [tex]\( f(2) = 4 \)[/tex]



Answer :

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.