Question

Using the included equations, compare the domain and range of the square root and cube root functions.

f(x)=2√x and g(x)=2^3√x

Responses

Both functions have the same Domain and Range: (-∞, ∞)

f(x) Domain is [0, ∞) and Range is [0, ∞). g(x) Domain is (-∞, ∞) and Range is (-∞, ∞)

g(x) Domain is [0, ∞) and Range is [0, ∞). f(x) Domain is (-∞, ∞) and Range is (-∞, ∞)

Answer 1

To compare the domain and range of the functions \( f(x) = 2\sqrt{x} \) and \( g(x) = 2\sqrt[3]{x} \), we can analyze each function.

### Function \( f(x) = 2\sqrt{x} \)
1. **Domain**: The square root function is only defined for non-negative inputs. Therefore, the domain of \( f(x) \) is:
\[
[0, \infty)
\]

2. **Range**: The output of the square root function is also non-negative. Thus, the range of \( f(x) \) is:
\[
[0, \infty)
\]

### Function \( g(x) = 2\sqrt[3]{x} \)
1. **Domain**: The cube root function is defined for all real numbers. Therefore, the domain of \( g(x) \) is:
\[
(-\infty, \infty)
\]

2. **Range**: The cube root function can produce all real numbers as output, thus the range of \( g(x) \) is:
\[
(-\infty, \infty)
\]

### Summary of Domain and Range
- \( f(x) = 2\sqrt{x} \):
- Domain: \([0, \infty)\)
- Range: \([0, \infty)\)

- \( g(x) = 2\sqrt[3]{x} \):
- Domain: \((-\infty, \infty)\)
- Range: \((-\infty, \infty)\)

### Conclusion
The correct response is:
**f(x) Domain is [0, ∞) and Range is [0, ∞). g(x) Domain is (-∞, ∞) and Range is (-∞, ∞)**.