Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Our platform provides a seamless experience for finding reliable answers from a knowledgeable network of professionals. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
To determine which function has the given properties, let's analyze each function step by step:
### Given Properties:
1. The domain is the set of all real numbers.
2. One [tex]\( x \)[/tex]-intercept is [tex]\( \left(\frac{\pi}{2}, 0\right) \)[/tex].
3. The maximum value is 3.
4. The [tex]\( y \)[/tex]-intercept is [tex]\( (0, -3) \)[/tex].
We will evaluate each function based on these properties.
### Analysis of Each Function
1. [tex]\( y = -3 \sin(x) \)[/tex]:
- Domain: The domain of sine function is all real numbers. Thus, this property is satisfied.
- [tex]\( x \)[/tex]-intercept: For [tex]\( y = -3 \sin(x) \)[/tex] to have an [tex]\( x \)[/tex]-intercept at [tex]\( \left(\frac{\pi}{2}, 0\right) \)[/tex]:
[tex]\[ -3 \sin\left(\frac{\pi}{2}\right) = -3 \cdot 1 = -3 \neq 0 \][/tex]
This property is not satisfied.
- Maximum value: The maximum value of [tex]\( -3 \sin(x) \)[/tex] is indeed 3 in magnitude but negative, so it does not satisfy this property.
- [tex]\( y \)[/tex]-intercept: At [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -3 \sin(0) = 0 \quad (\text{not } -3) \][/tex]
This property is not satisfied.
2. [tex]\( y = -3 \cos(x) \)[/tex]:
- Domain: The domain of cosine function is all real numbers. Thus, this property is satisfied.
- [tex]\( x \)[/tex]-intercept: For [tex]\( y = -3 \cos(x) \)[/tex] to have an [tex]\( x \)[/tex]-intercept at [tex]\( \left(\frac{\pi}{2}, 0\right) \)[/tex]:
[tex]\[ -3 \cos\left(\frac{\pi}{2}\right) = -3 \cdot 0 = 0 \][/tex]
This property is satisfied.
- Maximum value: The maximum value of [tex]\( -3 \cos(x) \)[/tex] is 0 in magnitude but negative, so the 3 as the maximum is not satisfied.
- [tex]\( y \)[/tex]-intercept: At [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -3 \cos(0) = -3 \][/tex]
This property is satisfied.
3. [tex]\( y = 3 \sin(x) \)[/tex]:
- Domain: The domain is again all real numbers. Thus, this property is satisfied.
- [tex]\( x \)[/tex]-intercept: For [tex]\( y = 3 \sin(x) \)[/tex] to have an [tex]\( x \)[/tex]-intercept at [tex]\( \left(\frac{\pi}{2}, 0\right) \)[/tex]:
[tex]\[ 3 \sin\left(\frac{\pi}{2}\right) = 3 \cdot 1 = 3 \][/tex]
This property is not satisfied.
- Maximum value: The maximum value of [tex]\( 3 \sin(x) \)[/tex] is 3.
- [tex]\( y \)[/tex]-intercept: At [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 3 \sin(0) = 0 \quad (\text{not } -3) \][/tex]
This property is not satisfied.
4. [tex]\( y = 3 \cos(x) \)[/tex]:
- Domain: The domain is all real numbers. Thus, this property is satisfied.
- [tex]\( x \)[/tex]-intercept: For [tex]\( y = 3 \cos(x) \)[/tex] to have an [tex]\( x \)[/tex]-intercept at [tex]\( \left(\frac{\pi}{2}, 0\right) \)[/tex]:
[tex]\[ 3 \cos\left(\frac{\pi}{2}\right) = 3 \cdot 0 = 0 \][/tex]
This property is satisfied.
- Maximum value: The maximum value of [tex]\( 3 \cos(x) \)[/tex] is:
[tex]\[ 3 \cos(0) = 3 \][/tex]
This property is satisfied.
- [tex]\( y \)[/tex]-intercept: At [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 3 \cos(0) = 3 \quad (\text{not } -3) \][/tex]
This property is not satisfied.
After analyzing all options, none of the functions fully satisfy all given properties. Therefore, the answer is:
[tex]\[ \boxed{0} \][/tex]
### Given Properties:
1. The domain is the set of all real numbers.
2. One [tex]\( x \)[/tex]-intercept is [tex]\( \left(\frac{\pi}{2}, 0\right) \)[/tex].
3. The maximum value is 3.
4. The [tex]\( y \)[/tex]-intercept is [tex]\( (0, -3) \)[/tex].
We will evaluate each function based on these properties.
### Analysis of Each Function
1. [tex]\( y = -3 \sin(x) \)[/tex]:
- Domain: The domain of sine function is all real numbers. Thus, this property is satisfied.
- [tex]\( x \)[/tex]-intercept: For [tex]\( y = -3 \sin(x) \)[/tex] to have an [tex]\( x \)[/tex]-intercept at [tex]\( \left(\frac{\pi}{2}, 0\right) \)[/tex]:
[tex]\[ -3 \sin\left(\frac{\pi}{2}\right) = -3 \cdot 1 = -3 \neq 0 \][/tex]
This property is not satisfied.
- Maximum value: The maximum value of [tex]\( -3 \sin(x) \)[/tex] is indeed 3 in magnitude but negative, so it does not satisfy this property.
- [tex]\( y \)[/tex]-intercept: At [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -3 \sin(0) = 0 \quad (\text{not } -3) \][/tex]
This property is not satisfied.
2. [tex]\( y = -3 \cos(x) \)[/tex]:
- Domain: The domain of cosine function is all real numbers. Thus, this property is satisfied.
- [tex]\( x \)[/tex]-intercept: For [tex]\( y = -3 \cos(x) \)[/tex] to have an [tex]\( x \)[/tex]-intercept at [tex]\( \left(\frac{\pi}{2}, 0\right) \)[/tex]:
[tex]\[ -3 \cos\left(\frac{\pi}{2}\right) = -3 \cdot 0 = 0 \][/tex]
This property is satisfied.
- Maximum value: The maximum value of [tex]\( -3 \cos(x) \)[/tex] is 0 in magnitude but negative, so the 3 as the maximum is not satisfied.
- [tex]\( y \)[/tex]-intercept: At [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -3 \cos(0) = -3 \][/tex]
This property is satisfied.
3. [tex]\( y = 3 \sin(x) \)[/tex]:
- Domain: The domain is again all real numbers. Thus, this property is satisfied.
- [tex]\( x \)[/tex]-intercept: For [tex]\( y = 3 \sin(x) \)[/tex] to have an [tex]\( x \)[/tex]-intercept at [tex]\( \left(\frac{\pi}{2}, 0\right) \)[/tex]:
[tex]\[ 3 \sin\left(\frac{\pi}{2}\right) = 3 \cdot 1 = 3 \][/tex]
This property is not satisfied.
- Maximum value: The maximum value of [tex]\( 3 \sin(x) \)[/tex] is 3.
- [tex]\( y \)[/tex]-intercept: At [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 3 \sin(0) = 0 \quad (\text{not } -3) \][/tex]
This property is not satisfied.
4. [tex]\( y = 3 \cos(x) \)[/tex]:
- Domain: The domain is all real numbers. Thus, this property is satisfied.
- [tex]\( x \)[/tex]-intercept: For [tex]\( y = 3 \cos(x) \)[/tex] to have an [tex]\( x \)[/tex]-intercept at [tex]\( \left(\frac{\pi}{2}, 0\right) \)[/tex]:
[tex]\[ 3 \cos\left(\frac{\pi}{2}\right) = 3 \cdot 0 = 0 \][/tex]
This property is satisfied.
- Maximum value: The maximum value of [tex]\( 3 \cos(x) \)[/tex] is:
[tex]\[ 3 \cos(0) = 3 \][/tex]
This property is satisfied.
- [tex]\( y \)[/tex]-intercept: At [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 3 \cos(0) = 3 \quad (\text{not } -3) \][/tex]
This property is not satisfied.
After analyzing all options, none of the functions fully satisfy all given properties. Therefore, the answer is:
[tex]\[ \boxed{0} \][/tex]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
