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 the same domain. 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)=4x−−√
and g(x)=4x−−√3
. Which of the options describing the graphs is true?

(1 point)
Responses

Option #3
Option #3

Option #4
Option #4

Option #2
Option #2

Option #1
Option #1
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Answer 1

To compare the graphs of \( f(x) = \sqrt{4x} \) and \( g(x) = \sqrt[3]{4x} \), we should analyze their domains and ranges.

1. Domain:

   • The function \( f(x) = \sqrt{4x} \) is defined for \( x \geq 0 \) (it only takes non-negative inputs because you cannot take the square root of a negative number).
   • The function \( g(x) = \sqrt[3]{4x} \) is defined for all real numbers \( x \) (the cube root function exists for both positive and negative inputs).

   Therefore, the functions have different domains.

2. Range:

   • The range of \( f(x) = \sqrt{4x} \) is \( [0, \infty) \) because it outputs non-negative values.
   • The range of \( g(x) = \sqrt[3]{4x} \) is \( (-\infty, \infty) \) because the cube root can produce any real number.

   Thus, the functions have different ranges.

3. Monotonicity:

   • The function \( f(x) = \sqrt{4x} \) is increasing as \( x \) increases.
   • The function \( g(x) = \sqrt[3]{4x} \) is also increasing as \( x \) increases.

   Both functions are increasing.

Based on the analysis:

• The functions have different domains: True
• The functions have different ranges: True
• Both functions are increasing: True

Thus, the correct option that describes the graphs of \( f(x) \) and \( g(x) \) is Option #3 (The functions have different domains; The functions have different ranges; Both functions are increasing).