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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]
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