the function f(x) = √(-x) is shown on the graph. which statement is correct? the domain of the function is…

Explanation:

Step1: Determine the domain of the function.

The function is $f(x) = \sqrt{-x}$. For the square root to be defined for real numbers, the expression under the radical must be non-negative. $$-x \ge 0$$

Step2: Solve the inequality for the domain.

Multiply the inequality by -1 and reverse the inequality sign. $$x \le 0$$ So, the domain is all real numbers less than or equal to 0, which is $(-\infty, 0]$.

Step3: Determine the range of the function.

The principal square root function, $\sqrt{u}$, always returns a non-negative value, where $u \ge 0$. In this case, $u = -x$. $$f(x) = \sqrt{-x} \ge 0$$ So, the range is all real numbers greater than or equal to 0, which is $[0, \infty)$.

Step4: Evaluate the given statements.

Compare the calculated domain and range with the options provided. Option 1: Domain $x \ge 0$ (Incorrect). Option 2: Range $y \ge -1$ (Incorrect, the range is $y \ge 0$). Option 3: Range $y \le 0$ (Incorrect). Option 4: Domain $x \le 0$ (Correct).

Answer:

The domain of the function is all real numbers less than or equal to 0.