Use the table to answer the question.

Question

Option #1 Option #2 Option #3 Option #4
The functions have the same domain. The functions have different domains. The functions have different domains. The functions have different domains.
The functions have the same range. The functions have different ranges. The functions have the same range. The functions have different ranges.
Both functions are decreasing. Both functions are decreasing. Both functions are increasing. Both functions are increasing.
Compare the graphs of f(x)=−2x−−√
and g(x)=−2x−−√3
. Which of the options describing the graphs is true?(1 point)
Responses

Option #4
Option #4

Option #1
Option #1

Option #2
Option #2

Option #3

Answer 1

To analyze the functions \( f(x) = -2\sqrt{x} \) and \( g(x) = -2\sqrt[3]{x} \), we can look at their domain and range, as well as their behavior (increasing or decreasing).

1. Domain:

   • f(x) = -2√x: The domain is \( x \geq 0 \), as square roots are only defined for non-negative numbers.
   • g(x) = -2∛x: The domain is all real numbers, \( x ∈ \mathbb{R} \), as cube roots are defined for all real numbers.

   Therefore, the functions have different domains.

2. Range:

   • f(x): As \( x \) increases from 0 to ∞, \( f(x) \) decreases from 0 to -∞. Hence, the range is \( (-\infty, 0] \).
   • g(x): The function can take any real value as \( x \) takes on values from -∞ to +∞. So, the range is \( (-\infty, 0) \) because it will never reach 0 (as it decreases as \( x \) increases).

   Hence, the functions also have different ranges.

3. Behavior (increasing or decreasing):

   • Both functions decrease as \( x \) increases in their respective domains.

Based on the analysis:

• The functions have different domains.
• The functions have different ranges.
• Both functions are decreasing.

Thus, the correct option that describes the graphs is Option #2:

• The functions have different domains.
• The functions have different ranges.
• Both functions are decreasing.