From the image you uploaded, the graph appears to represent a piecewise function $f(x)$ where the value of the function changes depending on the value of $x$. The goal is to fill in the conditions for each value of $f(x)$.

Analyzing the Graph

• For $f(x) = 5$:

  • There is a horizontal line at $y = 5$ from $x = -6$ to $x = -2$. This means $f(x) = 5$ when $-6 \leq x \leq -2$.

• For $f(x) = -1$:

  • There is another horizontal line at $y = -1$ between $x = 0$ and $x = 2$, but excluding $x = 0$ (open circle at $x = 0$) and including $x = 2$ (closed circle at $x = 2$). This means $f(x) = -1$ when $0 < x \leq 2$.

• For $f(x) = 3$:

  • There is a single point at $(4, 3)$, which means $f(x) = 3$ when $x = 4$.

Solution:

The piecewise function $f(x)$ can be written as: