Question

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

(1 point)
Responses

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 .

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.

Answer 1

To compare the graphs of \( f(x) = 2\sqrt{x} \) and \( g(x) = 2\sqrt[3]{x} \), let's analyze each statement.

1. **They are both decreasing on their domains.**
- For \( f(x) = 2\sqrt{x} \), the function is increasing on its domain \( x \geq 0 \).
- For \( g(x) = 2\sqrt[3]{x} \), the function is also increasing for all real \( x \).
- Therefore, this statement is **false**.

2. **Both graphs go through (−1,−1), (0,0), and (1,1).**
- For \( f(x) = 2\sqrt{x} \), it is only defined for \( x \geq 0 \) (so it does not go through \((-1, -1)\)).
- For \( g(x) = 2\sqrt[3]{x} \), \( g(-1) = 2\sqrt[3]{-1} = 2(-1) = -2 \) (so it does not go through \((-1, -1)\)).
- Therefore, this statement is **false**.

3. **When \( x > 1 \), the function \( g(x) = 2\sqrt[3]{x} \) is greater than \( f(x) = 2\sqrt{x} \).**
- For \( x > 1 \):
- \( f(x) = 2\sqrt{x} \) grows faster than \( g(x) = 2\sqrt[3]{x} \).
- Specifically, if you calculate values, e.g., for \( x = 2 \):
- \( f(2) = 2\sqrt{2} \approx 2.828 \),
- \( g(2) = 2\sqrt[3]{2} \approx 1.587 \).
- Thus, \( f(x) > g(x) \) when \( x > 1 \).
- Therefore, this statement is **false**.

4. **The graphs have different domains.**
- \( f(x) = 2\sqrt{x} \) has the domain \( [0, \infty) \).
- \( g(x) = 2\sqrt[3]{x} \) has the domain \( (-\infty, \infty) \).
- Since their domains are indeed different, this statement is **true**.

In conclusion, the true statement is:

**The graphs have different domains.**