Question
Compare the graphs of f(x)=x−−√+1 and g(x)=x−−√3+1 . Which one of the following options is true? (1 point) Responses Both functions have the same range, (−∞,∞) . Both functions have the same range, left parenthesis negative infinity comma infinity right parenthesis . Both functions go through the points (−1,0) and (0,1) . Both functions go through the points left parenthesis negative 1 comma 0 right parenthesis and left parenthesis 0 comma 1 right parenthesis . Both functions have the same domain, x>0 . Both functions have the same domain, x greater than 0 . Both functions are increasing on their domain.
Answers
GPT-4o mini
Let's analyze the functions \( f(x) = \sqrt{x} + 1 \) and \( g(x) = \sqrt[3]{x} + 1 \) to determine their characteristics.
### 1. Domain:
- **For \( f(x) \)**:
- The square root function \( \sqrt{x} \) is defined for \( x \geq 0 \).
- Thus, the domain of \( f(x) \) is \( [0, \infty) \).
- **For \( g(x) \)**:
- The cube root function \( \sqrt[3]{x} \) is defined for all real numbers.
- Thus, the domain of \( g(x) \) is \( (-\infty, \infty) \).
### 2. Range:
- **For \( f(x) \)**:
- As \( x \) approaches 0, \( f(0) = \sqrt{0} + 1 = 1 \).
- As \( x \) increases towards infinity, \( f(x) \) also increases towards infinity.
- This means the range of \( f(x) \) is \( [1, \infty) \).
- **For \( g(x) \)**:
- As \( x \) approaches negative infinity, \( g(x) = \sqrt[3]{x} + 1 \) approaches negative infinity.
- As \( x \) approaches positive infinity, \( g(x) \) approaches positive infinity.
- Therefore, the range of \( g(x) \) is \( (-\infty, \infty) \).
### 3. Points:
- **At \( x = -1 \)**:
- \( f(-1) \) is not defined because the square root function is not defined for negative values.
- \( g(-1) = \sqrt[3]{-1} + 1 = -1 + 1 = 0 \).
- **At \( x = 0 \)**:
- \( f(0) = \sqrt{0} + 1 = 1 \).
- \( g(0) = \sqrt[3]{0} + 1 = 0 + 1 = 1 \).
### 4. Monotonicity:
Both functions are increasing on their respective domains:
- \( f(x) \) is increasing for \( x \geq 0 \).
- \( g(x) \) is increasing for all \( x \).
### Summary:
Now, let's evaluate the options:
1. **Both functions have the same range, (−∞,∞)**: **False**
- \( f(x) \) has range \( [1, \infty) \) and \( g(x) \) has range \( (-\infty, \infty) \).
2. **Both functions go through the points (−1,0) and (0,1)**: **False**
- \( f(x) \) is not defined at \( x = -1 \), but both functions do go through (0, 1).
3. **Both functions have the same domain, x>0**: **False**
- \( f(x) \) is defined for \( [0, \infty) \) while \( g(x) \) is defined for all \( x \).
4. **Both functions are increasing on their domain**: **True**
- Both functions are indeed increasing over their respective domains.
### Conclusion:
The true statement is that **both functions are increasing on their domain**.
### 1. Domain:
- **For \( f(x) \)**:
- The square root function \( \sqrt{x} \) is defined for \( x \geq 0 \).
- Thus, the domain of \( f(x) \) is \( [0, \infty) \).
- **For \( g(x) \)**:
- The cube root function \( \sqrt[3]{x} \) is defined for all real numbers.
- Thus, the domain of \( g(x) \) is \( (-\infty, \infty) \).
### 2. Range:
- **For \( f(x) \)**:
- As \( x \) approaches 0, \( f(0) = \sqrt{0} + 1 = 1 \).
- As \( x \) increases towards infinity, \( f(x) \) also increases towards infinity.
- This means the range of \( f(x) \) is \( [1, \infty) \).
- **For \( g(x) \)**:
- As \( x \) approaches negative infinity, \( g(x) = \sqrt[3]{x} + 1 \) approaches negative infinity.
- As \( x \) approaches positive infinity, \( g(x) \) approaches positive infinity.
- Therefore, the range of \( g(x) \) is \( (-\infty, \infty) \).
### 3. Points:
- **At \( x = -1 \)**:
- \( f(-1) \) is not defined because the square root function is not defined for negative values.
- \( g(-1) = \sqrt[3]{-1} + 1 = -1 + 1 = 0 \).
- **At \( x = 0 \)**:
- \( f(0) = \sqrt{0} + 1 = 1 \).
- \( g(0) = \sqrt[3]{0} + 1 = 0 + 1 = 1 \).
### 4. Monotonicity:
Both functions are increasing on their respective domains:
- \( f(x) \) is increasing for \( x \geq 0 \).
- \( g(x) \) is increasing for all \( x \).
### Summary:
Now, let's evaluate the options:
1. **Both functions have the same range, (−∞,∞)**: **False**
- \( f(x) \) has range \( [1, \infty) \) and \( g(x) \) has range \( (-\infty, \infty) \).
2. **Both functions go through the points (−1,0) and (0,1)**: **False**
- \( f(x) \) is not defined at \( x = -1 \), but both functions do go through (0, 1).
3. **Both functions have the same domain, x>0**: **False**
- \( f(x) \) is defined for \( [0, \infty) \) while \( g(x) \) is defined for all \( x \).
4. **Both functions are increasing on their domain**: **True**
- Both functions are indeed increasing over their respective domains.
### Conclusion:
The true statement is that **both functions are increasing on their domain**.