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

Answer:

{f(x) | -∞ < f(x) ≤ 4}

Explanation:

Step1: Analyze the left part of the graph.

The left part of the graph is a horizontal ray starting from $x = -\infty$ and ending at $x=0$. The value of the function $f(x)$ on this part is constant and equal to 4. $$f(x) = 4 \quad \text{for} \quad x \le 0$$

Step2: Analyze the right part of the graph.

The right part of the graph is a ray starting from the point $(0, 4)$ and extending downwards to the right indefinitely. The y-values decrease from 4 as x increases. $$f(x) \le 4 \quad \text{for} \quad x > 0$$

Step3: Determine the overall range.

The range of a function is the set of all possible output values (y-values). Combining both parts, the maximum value of $f(x)$ is 4, and the function takes on all values less than or equal to 4. $$ \text{Range} = (-\infty, 4] $$

Step4: Express the range in set notation.

The range expressed in set notation is the set of all $f(x)$ such that $f(x)$ is less than or equal to 4. $$ {f(x) | -\infty < f(x) \le 4} $$ This corresponds to the second option.