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
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 .
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.
They are both decreasing on their domains.
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 .
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.
They are both decreasing on their domains.
Answers
GPT-4o mini
To analyze the functions \( f(x) = 2\sqrt{x} \) and \( g(x) = 2\sqrt[3]{x} \), let's break down the statements one by one and evaluate their validity:
1. **When \( x > 1 \), the function \( g(x) = 2\sqrt[3]{x} \) is greater than \( f(x) = 2\sqrt{x} \).**
- For \( x > 1 \):
- The function \( f(x) = 2\sqrt{x} \) grows faster than \( g(x) = 2\sqrt[3]{x} \) because the square root function increases more rapidly than the cube root function.
- So, for \( x > 1 \), \( f(x) > g(x) \).
- This statement is **false**.
2. **Both graphs go through \( (-1, -1) \), \( (0, 0) \), and \( (1, 1) \).**
- For \( f(x) \):
- \( f(-1) = 2\sqrt{-1} \) (undefined for real numbers)
- \( f(0) = 2\sqrt{0} = 0 \)
- \( f(1) = 2\sqrt{1} = 2 \)
- For \( g(x) \):
- \( g(-1) = 2\sqrt[3]{-1} = -2 \)
- \( g(0) = 2\sqrt[3]{0} = 0 \)
- \( g(1) = 2\sqrt[3]{1} = 2 \)
- Both functions do not go through \( (-1, -1) \) and have different outputs at \( x = -1 \). Hence, this statement is **false**.
3. **The graphs have different domains.**
- The domain of \( f(x) = 2\sqrt{x} \) is \( x \geq 0 \) (since the square root is not defined for negative numbers).
- The domain of \( g(x) = 2\sqrt[3]{x} \) is \( x \in \mathbb{R} \) (the cube root is defined for all real numbers).
- Therefore, they do have different domains. This statement is **true**.
4. **They are both decreasing on their domains.**
- The function \( f(x) = 2\sqrt{x} \) is increasing on its domain \( [0, \infty) \).
- The function \( g(x) = 2\sqrt[3]{x} \) is also increasing on its domain \( \mathbb{R} \).
- Therefore, this statement is **false**.
In conclusion, the only true statement from the options provided is:
**The graphs have different domains.**
1. **When \( x > 1 \), the function \( g(x) = 2\sqrt[3]{x} \) is greater than \( f(x) = 2\sqrt{x} \).**
- For \( x > 1 \):
- The function \( f(x) = 2\sqrt{x} \) grows faster than \( g(x) = 2\sqrt[3]{x} \) because the square root function increases more rapidly than the cube root function.
- So, for \( x > 1 \), \( f(x) > g(x) \).
- This statement is **false**.
2. **Both graphs go through \( (-1, -1) \), \( (0, 0) \), and \( (1, 1) \).**
- For \( f(x) \):
- \( f(-1) = 2\sqrt{-1} \) (undefined for real numbers)
- \( f(0) = 2\sqrt{0} = 0 \)
- \( f(1) = 2\sqrt{1} = 2 \)
- For \( g(x) \):
- \( g(-1) = 2\sqrt[3]{-1} = -2 \)
- \( g(0) = 2\sqrt[3]{0} = 0 \)
- \( g(1) = 2\sqrt[3]{1} = 2 \)
- Both functions do not go through \( (-1, -1) \) and have different outputs at \( x = -1 \). Hence, this statement is **false**.
3. **The graphs have different domains.**
- The domain of \( f(x) = 2\sqrt{x} \) is \( x \geq 0 \) (since the square root is not defined for negative numbers).
- The domain of \( g(x) = 2\sqrt[3]{x} \) is \( x \in \mathbb{R} \) (the cube root is defined for all real numbers).
- Therefore, they do have different domains. This statement is **true**.
4. **They are both decreasing on their domains.**
- The function \( f(x) = 2\sqrt{x} \) is increasing on its domain \( [0, \infty) \).
- The function \( g(x) = 2\sqrt[3]{x} \) is also increasing on its domain \( \mathbb{R} \).
- Therefore, this statement is **false**.
In conclusion, the only true statement from the options provided is:
**The graphs have different domains.**