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Sagot :
To determine which function has a domain of [tex]\((-\infty, \infty)\)[/tex] and a range of [tex]\((-\infty, 4]\)[/tex], we need to examine the domain and range of each of the given functions:
### Function 1: [tex]\( f(x) = -x^2 + 4 \)[/tex]
- Domain: The function [tex]\( f(x) = -x^2 + 4 \)[/tex] is a quadratic function and is defined for all real numbers [tex]\( x \)[/tex]. Therefore, its domain is [tex]\((-\infty, \infty)\)[/tex].
- Range: To determine the range, note that [tex]\( -x^2 + 4 \)[/tex] is a downward-opening parabola with a vertex at [tex]\((0, 4)\)[/tex]. This means that the maximum value is 4, and as [tex]\( x \)[/tex] moves away from 0, the value of the function decreases indefinitely.
Thus, the range of [tex]\( f(x) = -x^2 + 4 \)[/tex] is [tex]\((-\infty, 4]\)[/tex].
### Function 2: [tex]\( f(x) = 2^x + 4 \)[/tex]
- Domain: The exponential function [tex]\( 2^x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex], so [tex]\( 2^x + 4 \)[/tex] is also defined for all [tex]\( x \)[/tex]. Therefore, its domain is [tex]\((-\infty, \infty)\)[/tex].
- Range: Since [tex]\( 2^x \)[/tex] is always positive and increases asymptotically from 0 to [tex]\(\infty\)[/tex], [tex]\( 2^x + 4 \)[/tex] ranges from [tex]\( 4 \)[/tex] to [tex]\(\infty\)[/tex].
Thus, the range of [tex]\( f(x) = 2^x + 4 \)[/tex] is [tex]\((4, \infty)\)[/tex].
### Function 3: [tex]\( f(x) = x + 4 \)[/tex]
- Domain: The linear function [tex]\( x + 4 \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. Therefore, its domain is [tex]\((-\infty, \infty)\)[/tex].
- Range: Since a linear function [tex]\( x + 4 \)[/tex] can take any real value depending on [tex]\( x \)[/tex], its range is also all real numbers.
Thus, the range of [tex]\( f(x) = x + 4 \)[/tex] is [tex]\((-\infty, \infty)\)[/tex].
### Function 4: [tex]\( f(x) = -4x \)[/tex]
- Domain: The linear function [tex]\( -4x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. Therefore, its domain is [tex]\((-\infty, \infty)\)[/tex].
- Range: Similar to the previous linear function, [tex]\( -4x \)[/tex] can take any real value as [tex]\( x \)[/tex] can be any real number. Therefore, its range is also all real numbers.
Thus, the range of [tex]\( f(x) = -4x \)[/tex] is [tex]\((-\infty, \infty)\)[/tex].
### Conclusion
The function that satisfies both conditions, having a domain of [tex]\((-\infty, \infty)\)[/tex] and a range of [tex]\((-\infty, 4]\)[/tex], is:
[tex]\[ f(x) = -x^2 + 4 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
### Function 1: [tex]\( f(x) = -x^2 + 4 \)[/tex]
- Domain: The function [tex]\( f(x) = -x^2 + 4 \)[/tex] is a quadratic function and is defined for all real numbers [tex]\( x \)[/tex]. Therefore, its domain is [tex]\((-\infty, \infty)\)[/tex].
- Range: To determine the range, note that [tex]\( -x^2 + 4 \)[/tex] is a downward-opening parabola with a vertex at [tex]\((0, 4)\)[/tex]. This means that the maximum value is 4, and as [tex]\( x \)[/tex] moves away from 0, the value of the function decreases indefinitely.
Thus, the range of [tex]\( f(x) = -x^2 + 4 \)[/tex] is [tex]\((-\infty, 4]\)[/tex].
### Function 2: [tex]\( f(x) = 2^x + 4 \)[/tex]
- Domain: The exponential function [tex]\( 2^x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex], so [tex]\( 2^x + 4 \)[/tex] is also defined for all [tex]\( x \)[/tex]. Therefore, its domain is [tex]\((-\infty, \infty)\)[/tex].
- Range: Since [tex]\( 2^x \)[/tex] is always positive and increases asymptotically from 0 to [tex]\(\infty\)[/tex], [tex]\( 2^x + 4 \)[/tex] ranges from [tex]\( 4 \)[/tex] to [tex]\(\infty\)[/tex].
Thus, the range of [tex]\( f(x) = 2^x + 4 \)[/tex] is [tex]\((4, \infty)\)[/tex].
### Function 3: [tex]\( f(x) = x + 4 \)[/tex]
- Domain: The linear function [tex]\( x + 4 \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. Therefore, its domain is [tex]\((-\infty, \infty)\)[/tex].
- Range: Since a linear function [tex]\( x + 4 \)[/tex] can take any real value depending on [tex]\( x \)[/tex], its range is also all real numbers.
Thus, the range of [tex]\( f(x) = x + 4 \)[/tex] is [tex]\((-\infty, \infty)\)[/tex].
### Function 4: [tex]\( f(x) = -4x \)[/tex]
- Domain: The linear function [tex]\( -4x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. Therefore, its domain is [tex]\((-\infty, \infty)\)[/tex].
- Range: Similar to the previous linear function, [tex]\( -4x \)[/tex] can take any real value as [tex]\( x \)[/tex] can be any real number. Therefore, its range is also all real numbers.
Thus, the range of [tex]\( f(x) = -4x \)[/tex] is [tex]\((-\infty, \infty)\)[/tex].
### Conclusion
The function that satisfies both conditions, having a domain of [tex]\((-\infty, \infty)\)[/tex] and a range of [tex]\((-\infty, 4]\)[/tex], is:
[tex]\[ f(x) = -x^2 + 4 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
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