Question

Compare the graphs of f(x)=x−−√
and g(x)=x−−√3
. Which of the following options is true?(1 point)
Responses

Both functions have the same domain, [0,∞)
.
Both functions have the same domain, left bracket 0 comma infinity right parenthesis .

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 point (−1,−1)
.
Both functions go through the point left parenthesis negative 1 comma negative 1 right parenthesis .

Both functions are increasing.

Answer 1

Let’s analyze the two functions, \(f(x) = \sqrt{x}\) and \(g(x) = \sqrt[3]{x}\).

1. **Domain**:
- For \(f(x) = \sqrt{x}\), the domain is \( [0, \infty) \) because you can only take the square root of non-negative numbers.
- For \(g(x) = \sqrt[3]{x}\), the domain is \( (-\infty, \infty) \) since cube roots are defined for all real numbers.
- **Conclusion**: Both functions do **not** have the same domain.

2. **Range**:
- The range of \(f(x) = \sqrt{x}\) is \( [0, \infty) \) because the square root of any non-negative number is also non-negative.
- The range of \(g(x) = \sqrt[3]{x}\) is \( (-\infty, \infty) \) since a cube root can yield any real number.
- **Conclusion**: Both functions do **not** have the same range.

3. **Point (−1,−1)**:
- For \(f(x)\): \(f(-1) = \sqrt{-1}\) is not defined (it's imaginary).
- For \(g(x)\): \(g(-1) = \sqrt[3]{-1} = -1\).
- **Conclusion**: Both functions do **not** go through the point \((-1, -1)\).

4. **Increasing Behavior**:
- The function \(f(x) = \sqrt{x}\) is increasing on its domain \( [0, \infty) \).
- The function \(g(x) = \sqrt[3]{x}\) is also increasing on its entire domain \( (-\infty, \infty) \).
- **Conclusion**: Both functions are **increasing**.

Based on the above analysis, the true statement is:

**Both functions are increasing.**