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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 \(f\left(x\right)=\sqrt{4x}\) and \(g\left(x\right)=\sqrt[3]{4x}\). Which of the options describing the graphs is true?

(1 point)
Responses

Option #4
Option #4

Option #2
Option #2

Option #1
Option #1

Option #3
Option #3
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Answer 1

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

1. Domain:

   • For \( f(x) = \sqrt{4x} \): The expression inside the square root must be non-negative, so \( 4x \geq 0 \). This means \( x \geq 0 \). Therefore, the domain of \( f \) is \( [0, \infty) \).
   • For \( g(x) = \sqrt[3]{4x} \): The cube root is defined for all real numbers, so the domain of \( g \) is \( (-\infty, \infty) \).

   Conclusion: The functions have different domains.

2. Range:

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

   Conclusion: The functions have different ranges.

3. Increasing/Decreasing Behavior:

   • \( f(x) = \sqrt{4x} \) is an increasing function for \( x \geq 0 \) since its derivative \( f'(x) = \frac{2}{\sqrt{4x}} \) is positive for \( x > 0 \).
   • \( g(x) = \sqrt[3]{4x} \) is also an increasing function for all \( x \) since its derivative \( g'(x) = \frac{4}{3(4x)^{2/3}} \) is positive for all \( x \).

   Conclusion: Both functions are increasing.

Summary:

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

Given this analysis, the option that describes the functions accurately is:

Option #3: The functions have different domains, they have different ranges, and both functions are increasing.