Question

Determine which equations are the piecewise linear functions on the graphs.

Option #1: f(x)={−2x−1, x≤2−x+4, x>2

Option #2: f(x)={2x−1, x≥2x+4, x<2

Answer 1

To determine which of the given options represents a piecewise linear function, we need to evaluate the structure of each function.

**Option #1**:
\[
f(x) =
\begin{cases}
-2x - 1 & \text{if } x \leq 2 \\
-x + 4 & \text{if } x > 2
\end{cases}
\]

**Analysis**:
- The first part, \(-2x - 1\), is a linear function with a slope of -2.
- The second part, \(-x + 4\), is also a linear function with a slope of -1.
- Both parts are defined for specific ranges of \(x\) (less than or equal to 2 and greater than 2).

Since each piece is a linear function and they are joined at \(x = 2\), Option #1 represents a piecewise linear function.

**Option #2**:
\[
f(x) =
\begin{cases}
2x - 1 & \text{if } x \geq 2 \\
x + 4 & \text{if } x < 2
\end{cases}
\]

**Analysis**:
- The first part, \(2x - 1\), is a linear function with a slope of 2.
- The second part, \(x + 4\), is also a linear function with a slope of 1.
- Like in Option #1, both parts are linear functions defined over different ranges of \(x\) (greater than or equal to 2 and less than 2).

Thus, Option #2 also represents a piecewise linear function.

### Conclusion:
Both **Option #1** and **Option #2** are piecewise linear functions.