Use the image to answer the question. Compare the graphs of f(x)=−x−−√ and g(x)=−x−−√3 . Which of the following features about the graphs is true? (1 point) Responses The graphs have the same ranges. The graphs have the same ranges. The graphs are both increasing. The graphs are both increasing. The graphs are both decreasing. The graphs are both decreasing. The graphs have the same domains.

Question

GPT-4o mini · Answer

To analyze the functions $ f(x) = -\sqrt{x} $ and $ g(x) = -\sqrt[3]{x} $, let's break down the features of each graph. 1. **Domain**: - For $ f(x) = -\sqrt{x} $: The domain is $ x \geq 0 $ because you can only take the square root of non-negative numbers. - For $ g(x) = -\sqrt[3]{x} $: The domain is all real numbers $ x $ since the cube root is defined for all values. **Conclusion**: The graphs do NOT have the same domains. 2. **Range**: - For $ f(x) = -\sqrt{x} $: The range is $ y \leq 0 $ since the square root function outputs non-negative values and the negative sign flips it. - For $ g(x) = -\sqrt[3]{x} $: The range is also all real numbers $ y $ since the cube root function can produce both negative and positive values, and the negative sign flips the output. **Conclusion**: The graphs do NOT have the same ranges. 3. **Increasing/Decreasing**: - For $ f(x) = -\sqrt{x} $: This function is decreasing because as $ x $ increases, $ -\sqrt{x} $ becomes more negative. - For $ g(x) = -\sqrt[3]{x} $: This function also decreases because as $ x $ increases, $ -\sqrt[3]{x} $ becomes less negative (or goes up). **Conclusion**: Both graphs are decreasing. Given the options provided and the analysis above, the true statement is: **The graphs are both decreasing.**