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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

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 .

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

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.
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Answer 1

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

1. Points of Intersection:

   • For \( f(x) \):
     • When \( x = -1 \), \( f(-1) \) is not defined (since you cannot take the square root of a negative number).
     • When \( x = 0 \), \( f(0) = 2\sqrt{0} = 0 \).
     • When \( x = 1 \), \( f(1) = 2\sqrt{1} = 2 \).
   • For \( g(x) \):
     • When \( x = -1 \), \( g(-1) = 2\sqrt[3]{-1} = 2(-1) = -2 \).
     • When \( x = 0 \), \( g(0) = 2\sqrt[3]{0} = 0 \).
     • When \( x = 1 \), \( g(1) = 2\sqrt[3]{1} = 2(1) = 2 \).

   Therefore, both functions only go through (0,0) and (1,2), and they do not go through (-1, -1).

2. Monotonicity (Increasing/Decreasing):

   • The function \( f(x) = 2\sqrt{x} \) is only defined for \( x \geq 0 \) and is increasing on that interval.
   • The function \( g(x) = 2\sqrt[3]{x} \) is defined for all real numbers and is also increasing on its entire domain.

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

   • For \( x > 1 \), values of \( f(x) = 2\sqrt{x} \) and \( g(x) = 2\sqrt[3]{x} \) can be directly compared:

     • As \( x \) increases, \( f(x) \) will grow larger than \( g(x) \).
     • For example, when \( x = 2 \):
       • \( f(2) = 2\sqrt{2} \approx 2.83 \)
       • \( g(2) = 2\sqrt[3]{2} \approx 2.52 \)

     Hence, \( f(x) > g(x) \) for \( x > 1 \).

4. Domains:

   • The domain of \( f(x) \) is \( [0, \infty) \) (only non-negative numbers).
   • The domain of \( g(x) \) is \( (-\infty, \infty) \) (all real numbers).

From the above analysis, the correct statements are:

• They are both increasing on their domains.
• When \( x > 1 \), \( f(x) > g(x) \).
• The graphs have different domains.

Thus, the true statement from the options provided is:

The graphs have different domains.