Question

Using the included equations, compare domain and range of the linear and quadratic functions.

f(x)=(x+3)^2 and g(x) =2x+6

Responses:

Linear: Domain and Range are all real numbers. Quadratic: Domain is all real numbers and Range is all y values greater than or equal to 0

Both have the same Domain and Range: all real numbers

Linear: Domain: x>2 Range: y>6. Quadratic: Domain and Range: all real numbers

Answer 1

To compare the domain and range of the given functions \( f(x) = (x+3)^2 \) and \( g(x) = 2x + 6 \), we analyze each function separately.

### Function 1: \( f(x) = (x + 3)^2 \) (Quadratic Function)
- **Domain**: The function \( f(x) = (x+3)^2 \) is a polynomial (specifically, a quadratic), which means its domain includes all real numbers. In interval notation, this is expressed as:
\[
\text{Domain of } f: (-\infty, \infty)
\]

- **Range**: Since \( f(x) = (x+3)^2 \) represents a parabola that opens upwards, the minimum point occurs at the vertex, which is at \( x = -3 \). The value of \( f(-3) = 0 \). Therefore, the range of \( f(x) \) is all real numbers y such that \( y \) is greater than or equal to 0:
\[
\text{Range of } f: [0, \infty)
\]

### Function 2: \( g(x) = 2x + 6 \) (Linear Function)
- **Domain**: The function \( g(x) = 2x + 6 \) is a linear function, which means its domain also includes all real numbers:
\[
\text{Domain of } g: (-\infty, \infty)
\]

- **Range**: Since linear functions do not have any restrictions and extend infinitely in both directions, the range of \( g(x) \) is also all real numbers:
\[
\text{Range of } g: (-\infty, \infty)
\]

### Summary
- **Domain**:
- Both \( f(x) \) and \( g(x) \) have a domain of all real numbers: \((- \infty, \infty)\).

- **Range**:
- The range of \( f(x) \) is \( [0, \infty) \).
- The range of \( g(x) \) is \( (-\infty, \infty) \).

### Conclusion
The correct comparison is:
- **Linear**: Domain is all real numbers; Range is all real numbers.
- **Quadratic**: Domain is all real numbers; Range is all y values greater than or equal to 0.

Thus, the response that best fits this analysis is:
**Linear: Domain and Range are all real numbers. Quadratic: Domain is all real numbers and Range is all y values greater than or equal to 0.**