Question
A) Compare the graphs of f(x) = √x and g(x) =
3
√x. Which of the following options is true? (1 point)
Both functions have the same domain, [0, ∞).
Both functions are increasing.
Both functions have the same range, (−∞, ∞).
Both functions go through the point ( − 1, − 1).
3
√x. Which of the following options is true? (1 point)
Both functions have the same domain, [0, ∞).
Both functions are increasing.
Both functions have the same range, (−∞, ∞).
Both functions go through the point ( − 1, − 1).
Answers
GPT-4o mini
Let's analyze the two functions:
1. **Domain**:
- For \( f(x) = \sqrt{x} \), the domain is \( [0, \infty) \) because square roots are only defined for non-negative values of \( x \).
- For \( g(x) = \sqrt[3]{x} \) (which denotes the cube root), the domain is all real numbers \( (-\infty, \infty) \).
- **Conclusion**: The statement "Both functions have the same domain" is **false**.
2. **Increasing Behavior**:
- Both \( f(x) = \sqrt{x} \) and \( g(x) = \sqrt[3]{x} \) are increasing functions. The square root function increases as \( x \) increases, and the cube root function also increases as \( x \) increases.
- **Conclusion**: The statement "Both functions are increasing" is **true**.
3. **Range**:
- The range of \( f(x) = \sqrt{x} \) is \( [0, \infty) \) since it only outputs non-negative values.
- The range of \( g(x) = \sqrt[3]{x} \) is \( (-\infty, \infty) \) because cube roots can take on any real value.
- **Conclusion**: The statement "Both functions have the same range" is **false**.
4. **Point of Intersection**:
- The point \( (-1, -1) \) is not valid for \( f(x) = \sqrt{x} \) as square root of a negative number is not defined in the realm of real numbers. Therefore, \( f(-1) \) is undefined.
- The point \( (-1, -1) \) for \( g(x) = \sqrt[3]{-1} \) does yield \( -1 \), but since \( f(x) \) does not exist at \( -1 \), the point is not common.
- **Conclusion**: The statement "Both functions go through the point (−1, −1)" is **false**.
### Final Answer:
The true statement among the provided options is: **Both functions are increasing.**
1. **Domain**:
- For \( f(x) = \sqrt{x} \), the domain is \( [0, \infty) \) because square roots are only defined for non-negative values of \( x \).
- For \( g(x) = \sqrt[3]{x} \) (which denotes the cube root), the domain is all real numbers \( (-\infty, \infty) \).
- **Conclusion**: The statement "Both functions have the same domain" is **false**.
2. **Increasing Behavior**:
- Both \( f(x) = \sqrt{x} \) and \( g(x) = \sqrt[3]{x} \) are increasing functions. The square root function increases as \( x \) increases, and the cube root function also increases as \( x \) increases.
- **Conclusion**: The statement "Both functions are increasing" is **true**.
3. **Range**:
- The range of \( f(x) = \sqrt{x} \) is \( [0, \infty) \) since it only outputs non-negative values.
- The range of \( g(x) = \sqrt[3]{x} \) is \( (-\infty, \infty) \) because cube roots can take on any real value.
- **Conclusion**: The statement "Both functions have the same range" is **false**.
4. **Point of Intersection**:
- The point \( (-1, -1) \) is not valid for \( f(x) = \sqrt{x} \) as square root of a negative number is not defined in the realm of real numbers. Therefore, \( f(-1) \) is undefined.
- The point \( (-1, -1) \) for \( g(x) = \sqrt[3]{-1} \) does yield \( -1 \), but since \( f(x) \) does not exist at \( -1 \), the point is not common.
- **Conclusion**: The statement "Both functions go through the point (−1, −1)" is **false**.
### Final Answer:
The true statement among the provided options is: **Both functions are increasing.**