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To determine which function has a domain of [tex]\((-\infty, \infty)\)[/tex] and a range of [tex]\((-\infty, 4]\)[/tex], let's analyze each given function step-by-step.
1. [tex]\( f(x) = -x^2 + 4 \)[/tex]
- Domain: The domain of a function is the set of all possible input values (x-values) for the function. In the case of [tex]\( f(x) = -x^2 + 4 \)[/tex], [tex]\( x \)[/tex] can be any real number since there are no restrictions like square roots or denominators. Therefore, the domain is [tex]\((-\infty, \infty)\)[/tex].
- Range: The range is the set of all possible output values (y-values) for the function. The function [tex]\( f(x) = -x^2 + 4 \)[/tex] is a downward-facing parabola with the vertex at the point [tex]\((0, 4)\)[/tex]. The maximum value occurs at [tex]\( x = 0 \)[/tex], where [tex]\( f(0) = 4 \)[/tex]. As [tex]\( x \)[/tex] moves away from zero in either direction ([tex]\(\pm \infty\)[/tex]), [tex]\( f(x) \)[/tex] tends to [tex]\(-\infty\)[/tex]. Thus, the range of this function is [tex]\((-\infty, 4]\)[/tex].
2. [tex]\( f(x) = 2^x + 4 \)[/tex]
- Domain: For the exponential function [tex]\( 2^x \)[/tex], [tex]\( x \)[/tex] can be any real number, so the domain is [tex]\((-\infty, \infty)\)[/tex].
- Range: [tex]\( 2^x \)[/tex] is always positive for all real numbers [tex]\( x \)[/tex], and it approaches infinity as [tex]\( x \)[/tex] increases and approaches zero as [tex]\( x \)[/tex] decreases. Therefore, [tex]\( 2^x + 4 \)[/tex] shifts the range of [tex]\( 2^x \)[/tex] upwards by 4. Thus, the range of [tex]\( 2^x + 4 \)[/tex] is [tex]\((4, \infty)\)[/tex].
3. [tex]\( f(x) = x + 4 \)[/tex]
- Domain: The domain of a linear function [tex]\( f(x) = x + 4 \)[/tex] is all real numbers, so it is [tex]\((-\infty, \infty)\)[/tex].
- Range: The range of a linear function is also all real numbers, so it is [tex]\((-\infty, \infty)\)[/tex].
4. [tex]\( f(x) = -4x \)[/tex]
- Domain: The domain of the linear function [tex]\( f(x) = -4x \)[/tex] is all real numbers [tex]\((-\infty, \infty)\)[/tex].
- Range: The range of the linear function [tex]\( f(x) = -4x \)[/tex] is all real numbers [tex]\((-\infty, \infty)\)[/tex].
Considering all of the above functions, only [tex]\( f(x) = -x^2 + 4 \)[/tex] has both a domain of [tex]\((-\infty, \infty)\)[/tex] and a range of [tex]\((-\infty, 4]\)[/tex].
Thus, the correct answer is:
[tex]\[ f(x) = -x^2 + 4 \][/tex]
1. [tex]\( f(x) = -x^2 + 4 \)[/tex]
- Domain: The domain of a function is the set of all possible input values (x-values) for the function. In the case of [tex]\( f(x) = -x^2 + 4 \)[/tex], [tex]\( x \)[/tex] can be any real number since there are no restrictions like square roots or denominators. Therefore, the domain is [tex]\((-\infty, \infty)\)[/tex].
- Range: The range is the set of all possible output values (y-values) for the function. The function [tex]\( f(x) = -x^2 + 4 \)[/tex] is a downward-facing parabola with the vertex at the point [tex]\((0, 4)\)[/tex]. The maximum value occurs at [tex]\( x = 0 \)[/tex], where [tex]\( f(0) = 4 \)[/tex]. As [tex]\( x \)[/tex] moves away from zero in either direction ([tex]\(\pm \infty\)[/tex]), [tex]\( f(x) \)[/tex] tends to [tex]\(-\infty\)[/tex]. Thus, the range of this function is [tex]\((-\infty, 4]\)[/tex].
2. [tex]\( f(x) = 2^x + 4 \)[/tex]
- Domain: For the exponential function [tex]\( 2^x \)[/tex], [tex]\( x \)[/tex] can be any real number, so the domain is [tex]\((-\infty, \infty)\)[/tex].
- Range: [tex]\( 2^x \)[/tex] is always positive for all real numbers [tex]\( x \)[/tex], and it approaches infinity as [tex]\( x \)[/tex] increases and approaches zero as [tex]\( x \)[/tex] decreases. Therefore, [tex]\( 2^x + 4 \)[/tex] shifts the range of [tex]\( 2^x \)[/tex] upwards by 4. Thus, the range of [tex]\( 2^x + 4 \)[/tex] is [tex]\((4, \infty)\)[/tex].
3. [tex]\( f(x) = x + 4 \)[/tex]
- Domain: The domain of a linear function [tex]\( f(x) = x + 4 \)[/tex] is all real numbers, so it is [tex]\((-\infty, \infty)\)[/tex].
- Range: The range of a linear function is also all real numbers, so it is [tex]\((-\infty, \infty)\)[/tex].
4. [tex]\( f(x) = -4x \)[/tex]
- Domain: The domain of the linear function [tex]\( f(x) = -4x \)[/tex] is all real numbers [tex]\((-\infty, \infty)\)[/tex].
- Range: The range of the linear function [tex]\( f(x) = -4x \)[/tex] is all real numbers [tex]\((-\infty, \infty)\)[/tex].
Considering all of the above functions, only [tex]\( f(x) = -x^2 + 4 \)[/tex] has both a domain of [tex]\((-\infty, \infty)\)[/tex] and a range of [tex]\((-\infty, 4]\)[/tex].
Thus, the correct answer is:
[tex]\[ f(x) = -x^2 + 4 \][/tex]
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