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

Question

(1 point)
Responses

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 .

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.

Answer 1

Let's analyze the functions \( f(x) = 2\sqrt{x} \) and \( g(x) = 2\sqrt[3]{x} \) to determine which of the given statements is true.

1. Both graphs go through (-1,-1), (0,0), and (1,1):

   • \( f(-1) \) is not defined since square root of a negative number is not a real number. Thus, this statement is false.
   • \( f(0) = 2\sqrt{0} = 0 \) and \( g(0) = 2\sqrt[3]{0} = 0 \), so they both go through (0,0).
   • \( f(1) = 2\sqrt{1} = 2 \) and \( g(1) = 2\sqrt[3]{1} = 2 \), so they both go through (1,1).

   Therefore, this statement is false since both functions do not go through (-1,-1).

2. When \( x > 1 \), the function \( g(x) = 2\sqrt[3]{x} \) is greater than \( f(x) = 2\sqrt{x} \):

   • For \( x > 1 \), let's evaluate \( f(2) \) and \( g(2) \):
     • \( f(2) = 2\sqrt{2} \approx 2.828 \)
     • \( g(2) = 2\sqrt[3]{2} \approx 1.587 \)

   Since \( 2\sqrt{2} > 2\sqrt[3]{2} \), this statement is false.

3. The graphs have different domains:

   • The domain of \( f(x) = 2\sqrt{x} \) is \( x \geq 0 \) (only non-negative values).
   • The domain of \( g(x) = 2\sqrt[3]{x} \) is all real numbers (no restrictions).

   Therefore, this statement is true.

4. They are both decreasing on their domains:

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

   Therefore, this statement is false.

Conclusion:

The only true statement is: The graphs have different domains.