the graph of the piece - wise function f(x) is shown. what is the range of f(x)? o {f(x)|-∞ <f(x) < ∞} o…

Explanation:

Step1: Analyze the left piece of the graph.

The left part of the graph is a horizontal line segment where the y-value is constant at $f(x) = 4$. This piece covers the interval $x \in (-\infty, -1]$. The y-value attained is $y=4$.

Step2: Analyze the right piece of the graph.

The right part of the graph starts at the point $(-1, 4)$ (solid dot, so $y=4$ is included) and decreases linearly as $x$ increases. The line extends downwards indefinitely. Thus, this piece covers all y-values from $4$ down to $-\infty$. The y-values covered are $y \in (-\infty, 4]$.

Step3: Determine the overall range.

The range of the function is the union of all possible y-values from both pieces. The left piece gives $y=4$, and the right piece gives $y \le 4$. Combining these, the set of all possible y-values is $(-\infty, 4]$. In inequality notation, this is $-\infty < f(x) \le 4$.

Answer:

${f(x) | -\infty < f(x) \le 4}$