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Question

A coordinate plane's axes range from negative 4 to 4, both by 1-unit increments. A dotted curve labeled g of x and a solid curve labeled f of x are plotted.

Compare the given graphs of f(x)=2x−−√ and g(x)=2^3√x−− and determine which of the following is true.

(1 point)
Responses

When x>1, the function g(x)=2x−−√3 is greater than f(x)=2x−−√.
When x greater than 1 , the function g left parenthesis x right parenthesis equals 2 root index 3 Start Root x End Root is greater than f left parenthesis x right parenthesis equals 2 Start Root x End Root .

They are both decreasing on their domains.
They are both decreasing on their domains.

Both graphs go through (−1,−1), (0,0), and (1,1).
Both graphs go through left parenthesis negative 1 comma negative 1 right parenthesis , left parenthesis 0 comma 0 right parenthesis , and left parenthesis 1 comma 1 right parenthesis .

The graphs have different domains.
The graphs have different domains.
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Answer 1

To compare the functions \( f(x) = 2\sqrt{x} \) and \( g(x) = \sqrt[3]{2x} \), we need to analyze their domains, behavior, and points on the graphs.

1. Domains:

   • For \( f(x) = 2\sqrt{x} \), the function is defined for \( x \geq 0 \) because square roots of negative numbers are not defined in the real number system.
   • For \( g(x) = \sqrt[3]{2x} \), the function is defined for all \( x \) because cube roots are defined for all real numbers.

   This means that the two functions do indeed have different domains.

2. Behavior:

   • \( f(x) = 2\sqrt{x} \) is increasing for \( x \geq 0 \).
   • \( g(x) = \sqrt[3]{2x} \) is also increasing for all real \( x \).

   Thus, they are not both decreasing on their domains.

3. Points:

   • For \( f(x) \):
     • \( f(-1) \) is undefined.
     • \( f(0) = 0 \)
     • \( f(1) = 2 \)
   • For \( g(x) \):
     • \( g(-1) = \sqrt[3]{-2} \) (which is a negative value, around -1.26)
     • \( g(0) = 0 \)
     • \( g(1) = \sqrt[3]{2} \) (which is approximately 1.26)

   Therefore, both graphs do not pass through the same points \( (-1, -1) \), \( (0, 0) \), and \( (1, 1) \).

Based on the analysis, the correct statement is:

The graphs have different domains.