Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Get immediate and reliable solutions to your questions from a knowledgeable community of professionals on our platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
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]
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
