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)
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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: \[ \text{Domain of } f: (-\infty, \infty) \]
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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)
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Domain: The function \( g(x) = 2x + 6 \) is a linear function, which means its domain also includes all real numbers: \[ \text{Domain of } g: (-\infty, \infty) \]
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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
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Domain:
- Both \( f(x) \) and \( g(x) \) have a domain of all real numbers: \((- \infty, \infty)\).
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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.