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Sagot :
To determine which function could have the domain [tex]\((-\infty, \infty)\)[/tex] and the range [tex]\((-\infty, 2)\)[/tex], we will analyze the behavior of each function.
### Function A: [tex]\(y = -3^x + 2\)[/tex]
1. Domain: The exponential function [tex]\(3^x\)[/tex] is defined for all real numbers [tex]\(x\)[/tex]. Thus, the domain of [tex]\(y = -3^x + 2\)[/tex] is [tex]\((-\infty, \infty)\)[/tex].
2. Range: To determine the range, let's analyze the behavior of [tex]\(y\)[/tex].
- As [tex]\(x \to \infty\)[/tex], [tex]\(3^x \to \infty\)[/tex]. Therefore, [tex]\(-3^x \to -\infty\)[/tex] and hence, [tex]\(y = -3^x + 2 \to -\infty\)[/tex].
- As [tex]\(x \to -\infty\)[/tex], [tex]\(3^x \to 0\)[/tex]. Hence, [tex]\(-3^x \to 0\)[/tex] and [tex]\(y = 0 + 2 = 2\)[/tex].
Thus, the range is [tex]\((-\infty, 2)\)[/tex]. Therefore, function [tex]\(A\)[/tex] satisfies the given condition.
### Function B: [tex]\(v = -(0.25)^x - 2\)[/tex]
1. Domain: The exponential function [tex]\((0.25)^x\)[/tex] is defined for all real numbers [tex]\(x\)[/tex]. Thus, the domain of [tex]\(v = -(0.25)^x - 2\)[/tex] is [tex]\((-\infty, \infty)\)[/tex].
2. Range: Let's analyze the behavior of [tex]\(v\)[/tex].
- As [tex]\(x \to \infty\)[/tex], [tex]\((0.25)^x \to 0\)[/tex]. Therefore, [tex]\(-(0.25)^x \to 0\)[/tex] and [tex]\(v = 0 - 2 = -2\)[/tex].
- As [tex]\(x \to -\infty\)[/tex], [tex]\((0.25)^x \to \infty\)[/tex]. Hence, [tex]\(-(0.25)^x \to -\infty\)[/tex] and [tex]\(v \to -\infty\)[/tex].
Thus, the range is [tex]\((-\infty, -2)\)[/tex]. Therefore, function [tex]\(B\)[/tex] does not satisfy the given condition.
### Function C: [tex]\(y = -0.25^x + 2\)[/tex]
1. Domain: The exponential function [tex]\(0.25^x\)[/tex] is defined for all real numbers [tex]\(x\)[/tex]. Thus, the domain of [tex]\(y = -0.25^x + 2\)[/tex] is [tex]\((-\infty, \infty)\)[/tex].
2. Range: Let's analyze the behavior of [tex]\(y\)[/tex].
- As [tex]\(x \to \infty\)[/tex], [tex]\((0.25)^x \to 0\)[/tex]. Therefore, [tex]\(-0.25^x \to 0\)[/tex] and [tex]\(y = 0 + 2 = 2\)[/tex].
- As [tex]\(x \to -\infty\)[/tex], [tex]\((0.25)^x \to \infty\)[/tex]. Hence, [tex]\(-0.25^x \to -\infty\)[/tex] and [tex]\(y \to -\infty\)[/tex].
Thus, the range is [tex]\((-\infty, 2)\)[/tex]. Therefore, function [tex]\(C\)[/tex] satisfies the given condition.
### Function D: [tex]\(v = -3^x - 2\)[/tex]
1. Domain: The exponential function [tex]\(3^x\)[/tex] is defined for all real numbers [tex]\(x\)[/tex]. Thus, the domain of [tex]\(v = -3^x - 2\)[/tex] is [tex]\((-\infty, \infty)\)[/tex].
2. Range: Let's analyze the behavior of the function [tex]\(v\)[/tex]:
- As [tex]\(x \to \infty\)[/tex], [tex]\(3^x \to \infty\)[/tex]. Therefore, [tex]\(-3^x \to -\infty\)[/tex] and [tex]\(v = -3^x - 2 \to -\infty\)[/tex].
- As [tex]\(x \to -\infty\)[/tex], [tex]\(3^x \to 0\)[/tex]. Hence, [tex]\(-3^x \to 0\)[/tex], and [tex]\(v = 0 - 2 = -2\)[/tex].
Thus, the range is [tex]\((-\infty, -2)\)[/tex]. Therefore, function [tex]\(D\)[/tex] does not satisfy the given condition.
### Conclusion
The functions that have the domain [tex]\((-\infty, \infty)\)[/tex] and the range [tex]\((-\infty, 2)\)[/tex] are:
- [tex]\(A\)[/tex] : [tex]\(y = -3^x + 2\)[/tex]
- [tex]\(C\)[/tex] : [tex]\(y = -0.25^x + 2\)[/tex]
Thus, the correct options are:
- A. [tex]\(y = -3^x + 2\)[/tex]
- C. [tex]\(y = -0.25^x + 2\)[/tex]
### Function A: [tex]\(y = -3^x + 2\)[/tex]
1. Domain: The exponential function [tex]\(3^x\)[/tex] is defined for all real numbers [tex]\(x\)[/tex]. Thus, the domain of [tex]\(y = -3^x + 2\)[/tex] is [tex]\((-\infty, \infty)\)[/tex].
2. Range: To determine the range, let's analyze the behavior of [tex]\(y\)[/tex].
- As [tex]\(x \to \infty\)[/tex], [tex]\(3^x \to \infty\)[/tex]. Therefore, [tex]\(-3^x \to -\infty\)[/tex] and hence, [tex]\(y = -3^x + 2 \to -\infty\)[/tex].
- As [tex]\(x \to -\infty\)[/tex], [tex]\(3^x \to 0\)[/tex]. Hence, [tex]\(-3^x \to 0\)[/tex] and [tex]\(y = 0 + 2 = 2\)[/tex].
Thus, the range is [tex]\((-\infty, 2)\)[/tex]. Therefore, function [tex]\(A\)[/tex] satisfies the given condition.
### Function B: [tex]\(v = -(0.25)^x - 2\)[/tex]
1. Domain: The exponential function [tex]\((0.25)^x\)[/tex] is defined for all real numbers [tex]\(x\)[/tex]. Thus, the domain of [tex]\(v = -(0.25)^x - 2\)[/tex] is [tex]\((-\infty, \infty)\)[/tex].
2. Range: Let's analyze the behavior of [tex]\(v\)[/tex].
- As [tex]\(x \to \infty\)[/tex], [tex]\((0.25)^x \to 0\)[/tex]. Therefore, [tex]\(-(0.25)^x \to 0\)[/tex] and [tex]\(v = 0 - 2 = -2\)[/tex].
- As [tex]\(x \to -\infty\)[/tex], [tex]\((0.25)^x \to \infty\)[/tex]. Hence, [tex]\(-(0.25)^x \to -\infty\)[/tex] and [tex]\(v \to -\infty\)[/tex].
Thus, the range is [tex]\((-\infty, -2)\)[/tex]. Therefore, function [tex]\(B\)[/tex] does not satisfy the given condition.
### Function C: [tex]\(y = -0.25^x + 2\)[/tex]
1. Domain: The exponential function [tex]\(0.25^x\)[/tex] is defined for all real numbers [tex]\(x\)[/tex]. Thus, the domain of [tex]\(y = -0.25^x + 2\)[/tex] is [tex]\((-\infty, \infty)\)[/tex].
2. Range: Let's analyze the behavior of [tex]\(y\)[/tex].
- As [tex]\(x \to \infty\)[/tex], [tex]\((0.25)^x \to 0\)[/tex]. Therefore, [tex]\(-0.25^x \to 0\)[/tex] and [tex]\(y = 0 + 2 = 2\)[/tex].
- As [tex]\(x \to -\infty\)[/tex], [tex]\((0.25)^x \to \infty\)[/tex]. Hence, [tex]\(-0.25^x \to -\infty\)[/tex] and [tex]\(y \to -\infty\)[/tex].
Thus, the range is [tex]\((-\infty, 2)\)[/tex]. Therefore, function [tex]\(C\)[/tex] satisfies the given condition.
### Function D: [tex]\(v = -3^x - 2\)[/tex]
1. Domain: The exponential function [tex]\(3^x\)[/tex] is defined for all real numbers [tex]\(x\)[/tex]. Thus, the domain of [tex]\(v = -3^x - 2\)[/tex] is [tex]\((-\infty, \infty)\)[/tex].
2. Range: Let's analyze the behavior of the function [tex]\(v\)[/tex]:
- As [tex]\(x \to \infty\)[/tex], [tex]\(3^x \to \infty\)[/tex]. Therefore, [tex]\(-3^x \to -\infty\)[/tex] and [tex]\(v = -3^x - 2 \to -\infty\)[/tex].
- As [tex]\(x \to -\infty\)[/tex], [tex]\(3^x \to 0\)[/tex]. Hence, [tex]\(-3^x \to 0\)[/tex], and [tex]\(v = 0 - 2 = -2\)[/tex].
Thus, the range is [tex]\((-\infty, -2)\)[/tex]. Therefore, function [tex]\(D\)[/tex] does not satisfy the given condition.
### Conclusion
The functions that have the domain [tex]\((-\infty, \infty)\)[/tex] and the range [tex]\((-\infty, 2)\)[/tex] are:
- [tex]\(A\)[/tex] : [tex]\(y = -3^x + 2\)[/tex]
- [tex]\(C\)[/tex] : [tex]\(y = -0.25^x + 2\)[/tex]
Thus, the correct options are:
- A. [tex]\(y = -3^x + 2\)[/tex]
- C. [tex]\(y = -0.25^x + 2\)[/tex]
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