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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)=2x−−√3 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 .

The graphs have different domains.
The graphs have different domains.

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 .
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Answer 1

To compare the functions \( f(x) = 2\sqrt{x} \) and \( g(x) = 2\sqrt[3]{x} \), we can analyze the characteristics of both functions.

1. Domains:

   • The domain of \( f(x) = 2\sqrt{x} \) is \( x \geq 0 \) because the square root function is only defined for non-negative values.
   • The domain of \( g(x) = 2\sqrt[3]{x} \) is all real numbers \( (-\infty, \infty) \) because the cube root function is defined for all real numbers.

   Therefore, the graphs have different domains.

2. Behavior of the Functions:

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

   Thus, they are not decreasing on their domains.

3. Points:

   • Checking the points:
     • \( f(-1) \) is not defined.
     • \( f(0) = 0 \)
     • \( f(1) = 2\sqrt{1} = 2 \)
     • For \( g(x) \):
       • \( g(-1) = 2\sqrt[3]{-1} = -2 \)
       • \( g(0) = 0 \)
       • \( g(1) = 2\sqrt[3]{1} = 2 \)

   This shows that neither function goes through \((-1, -1)\), and they do not match at that point, while they both pass through \((0, 0)\) and \((1, 2)\).

4. Comparison for \( x > 1 \):

   • For \( x > 1 \), since \( \sqrt[3]{x} < \sqrt{x} \) for \( x > 1 \) (because the square root grows faster than the cube root), it follows that \( g(x) < f(x) \) for \( x > 1 \).

Therefore, the correct statements based on this analysis are:

• The graphs have different domains.
• They are both increasing on their domains.
• Both functions do not go through \((-1, -1)\) and indeed misrepresent the intersection point accusations.

Thus, among the options you've provided, the most fitting response would be:

The graphs have different domains.