On a coordinate plane, a piecewise function has 2 connecting lines. The first line starts at (0, 0) and goes up and to the left through (negative 5, 5) with an arrow instead of an endpoint. The second line starts at (0, 0) and goes up and to the right through (5, 5), with an arrow instead of an endpoint.

Question

The graph represents the piecewise function:
The image shows a piecewise function \( f(x) \) defined as follows:

\[
f(x) = \begin{cases}
x, & x > 0 \\
0, & x = 0 \\
-x, & x < 0
\end{cases}

What is the domain and range of the function?

Domain:
CHOOSE ONE
all integer numbers
all integer numbers greater than or equal to 0
all real numbers
all real numbers greater than or equal to 0

Range:
CHOOSE ONE
all integer numbers
all integer numbers greater than or equal to 0
all real numbers
all real numbers greater than or equal to 0

Answer 1

To analyze the piecewise function \( f(x) \):

1. Understanding the structure of the function:

   • For \( x > 0 \), \( f(x) = x \). This part of the function represents a line that rises to the right starting from the origin (0, 0).
   • For \( x = 0 \), \( f(x) = 0 \). This represents the point at the origin.
   • For \( x < 0 \), \( f(x) = -x \). This part of the function represents a line that rises to the left, starting at the origin as well.

2. Determining the domain: The domain of the function is the set of all possible input values \( x \) for which the function is defined:

   • Since \( f(x) \) is defined for all values \( x \) (negative, zero, and positive), the domain is all real numbers.

   Domain: all real numbers.

3. Determining the range: The range of the function is the set of all possible output values \( f(x) \):

   • When \( x > 0 \), \( f(x) = x \) means that \( f(x) \) takes on all positive real values.
   • When \( x = 0 \), \( f(0) = 0\), so 0 is included in the range.
   • When \( x < 0 \), \( f(x) = -x \) means that \( f(x) \) takes on all positive real values (since \( x \) is negative, \( -x \) becomes positive).

   Therefore, the function outputs values from 0 to positive infinity.

   Range: all real numbers greater than or equal to 0.

Final answers:

• Domain: all real numbers
• Range: all real numbers greater than or equal to 0