Question
Compare the graphs of f(x)=√(4x) and g(x)=∛(4x). Identify which of the following features is true about the graphs.
Answers
Answered by
GPT-4o mini
To compare the graphs of the functions \( f(x) = \sqrt{4x} \) and \( g(x) = \sqrt[3]{4x} \), let’s analyze their properties and characteristics:
1. **Domain**:
- \( f(x) = \sqrt{4x} \): The domain is \( x \geq 0 \) because the square root function is defined only for non-negative values.
- \( g(x) = \sqrt[3]{4x} \): The domain is all real numbers \( x \in \mathbb{R} \) since the cube root function is defined for all real numbers.
2. **Range**:
- \( f(x) \): The range is \( y \geq 0 \) because the square root function yields non-negative results.
- \( g(x) \): The range is all real numbers \( y \in \mathbb{R} \) because the cube root can produce both positive and negative outputs.
3. **Behavior at key points**:
- At \( x = 0 \):
- \( f(0) = \sqrt{0} = 0 \)
- \( g(0) = \sqrt[3]{0} = 0 \)
- As \( x \) approaches positive infinity (\( x \to \infty \)):
- \( f(x) \to \infty \)
- \( g(x) \to \infty \)
- As \( x \) approaches negative infinity (\( x \to -\infty \)):
- \( f(x) \) is not defined.
- \( g(x) \to -\infty \).
4. **Growth Rate**:
- For large positive \( x \):
- The function \( f(x) = \sqrt{4x} = 2\sqrt{x} \) grows slower than \( g(x) = \sqrt[3]{4x} = 2\sqrt[3]{x} \) since root functions with even degrees (like the square root) increase slower than root functions with odd degrees (like cube roots).
5. **Intercepts**:
- Both functions intercept the origin at the point (0, 0).
6. **End Behavior**:
- As \( x \to 0^+ \), \( f(x) \) approaches \( 0 \) more rapidly compared to the behavior of \( g(x) \).
Based on this analysis, key features we can conclude about the graphs of \( f(x) \) and \( g(x) \) are:
- \( f(x) \) is only defined for \( x \geq 0 \) while \( g(x) \) is defined for all \( x \).
- The range of \( f(x) \) is \( y \geq 0 \) whereas \( g(x) \) covers all real numbers.
- The growth rates differ because \( g(x) \) can produce negative outputs while \( f(x) \) cannot.
A specific statement that is true about the graphs would be:
- **The graph of \( f(x) \) only exists in the first quadrant (and the y-axis), while the graph of \( g(x) \) exists in all four quadrants.**
1. **Domain**:
- \( f(x) = \sqrt{4x} \): The domain is \( x \geq 0 \) because the square root function is defined only for non-negative values.
- \( g(x) = \sqrt[3]{4x} \): The domain is all real numbers \( x \in \mathbb{R} \) since the cube root function is defined for all real numbers.
2. **Range**:
- \( f(x) \): The range is \( y \geq 0 \) because the square root function yields non-negative results.
- \( g(x) \): The range is all real numbers \( y \in \mathbb{R} \) because the cube root can produce both positive and negative outputs.
3. **Behavior at key points**:
- At \( x = 0 \):
- \( f(0) = \sqrt{0} = 0 \)
- \( g(0) = \sqrt[3]{0} = 0 \)
- As \( x \) approaches positive infinity (\( x \to \infty \)):
- \( f(x) \to \infty \)
- \( g(x) \to \infty \)
- As \( x \) approaches negative infinity (\( x \to -\infty \)):
- \( f(x) \) is not defined.
- \( g(x) \to -\infty \).
4. **Growth Rate**:
- For large positive \( x \):
- The function \( f(x) = \sqrt{4x} = 2\sqrt{x} \) grows slower than \( g(x) = \sqrt[3]{4x} = 2\sqrt[3]{x} \) since root functions with even degrees (like the square root) increase slower than root functions with odd degrees (like cube roots).
5. **Intercepts**:
- Both functions intercept the origin at the point (0, 0).
6. **End Behavior**:
- As \( x \to 0^+ \), \( f(x) \) approaches \( 0 \) more rapidly compared to the behavior of \( g(x) \).
Based on this analysis, key features we can conclude about the graphs of \( f(x) \) and \( g(x) \) are:
- \( f(x) \) is only defined for \( x \geq 0 \) while \( g(x) \) is defined for all \( x \).
- The range of \( f(x) \) is \( y \geq 0 \) whereas \( g(x) \) covers all real numbers.
- The growth rates differ because \( g(x) \) can produce negative outputs while \( f(x) \) cannot.
A specific statement that is true about the graphs would be:
- **The graph of \( f(x) \) only exists in the first quadrant (and the y-axis), while the graph of \( g(x) \) exists in all four quadrants.**