Question

Question

Use the image to answer the question.

One solid and one dotted curve are drawn in a coordinate plane. The axes range from negative 2 to 2, both by 1-unit increments.

Compare the graphs of \(f\left(x\right)=-\sqrt{x}\) and \(g\left(x\right)=-\sqrt[3]{x}\). Which of the following features about the graphs is true?

(1 point)
Responses

The graphs are both increasing.
The graphs are both increasing.

The graphs have the same ranges.
The graphs have the same ranges.

The graphs have the same domains.
The graphs have the same domains.

The graphs are both decreasing.
The graphs are both decreasing.
Skip to navigation

Answer 1

To compare the functions \(f(x) = -\sqrt{x}\) and \(g(x) = -\sqrt[3]{x}\), we need to analyze their domains, ranges, and whether they are increasing or decreasing.

1. Domains:

   • \(f(x) = -\sqrt{x}\): The domain is \(x \geq 0\) since the square root function is defined only for non-negative values.
   • \(g(x) = -\sqrt[3]{x}\): The domain is all real numbers (\(-\infty < x < \infty\)) because the cube root function is defined for all real values.

   Conclusion: The graphs do not have the same domains.

2. Ranges:

   • For \(f(x)\): As \(x\) increases from 0 to \(\infty\), \(f(x)\) decreases from 0 to \(-\infty\). Therefore, the range of \(f(x)\) is \((-\infty, 0]\).
   • For \(g(x)\): Since \(g(x) = -\sqrt[3]{x}\) takes values from \(-\infty\) to 0 as \(x\) goes from \(-\infty\) to \(\infty\), the range of \(g(x)\) is also \((-\infty, 0]\).

   Conclusion: The graphs have the same ranges.

3. Increasing or Decreasing:

   • \(f(x) = -\sqrt{x}\): This function is decreasing because as \(x\) increases, \(-\sqrt{x}\) decreases.
   • \(g(x) = -\sqrt[3]{x}\): This function is also decreasing because as \(x\) increases, \(-\sqrt[3]{x}\) decreases.

   Conclusion: Both functions are decreasing.

Based on the analysis, the correct response is:

• The graphs have the same ranges.

Additionally, both functions are decreasing, so "The graphs are both decreasing" is also true. However, since the question only provides to select one feature, the answer is: The graphs have the same ranges.