Answer :
To determine which function matches the given properties, let's analyze each function with respect to the given criteria:
### Properties to check:
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].
### Functions to consider:
1. [tex]\( y = -3 \sin(x) \)[/tex]
2. [tex]\( y = -3 \cos(x) \)[/tex]
3. [tex]\( y = 3 \sin(x) \)[/tex]
4. [tex]\( y = 3 \cos(x) \)[/tex]
### Step-by-Step Analysis:
#### 1. [tex]\( y = -3 \sin(x) \)[/tex]
- Domain: The domain of [tex]\(\sin(x)\)[/tex] is all real numbers.
- [tex]\( x \)[/tex]-intercept: Solve [tex]\(-3 \sin(x) = 0\)[/tex]. This implies [tex]\(\sin(x) = 0\)[/tex], which occurs at [tex]\( x = n\pi \)[/tex] for integer [tex]\( n \)[/tex]. An [tex]\( x \)[/tex]-intercept at [tex]\(\left(\frac{\pi}{2}, 0\right)\)[/tex] does not occur.
- Maximum value: The maximum value of [tex]\(-3 \sin(x)\)[/tex] is [tex]\( -3 \)[/tex].
- [tex]\( y \)[/tex]-intercept: At [tex]\( x = 0 \)[/tex], [tex]\( y = -3 \sin(0) = 0 \)[/tex]. This does not match the given [tex]\( y \)[/tex]-intercept of [tex]\((0, -3)\)[/tex].
#### 2. [tex]\( y = -3 \cos(x) \)[/tex]
- Domain: The domain of [tex]\(\cos(x)\)[/tex] is all real numbers.
- [tex]\( x \)[/tex]-intercept: Solve [tex]\(-3 \cos(x) = 0\)[/tex]. This implies [tex]\(\cos(x) = 0\)[/tex], which occurs at [tex]\( x = \left(\frac{\pi}{2} + n\pi\right) \)[/tex] for integer [tex]\( n \)[/tex]. Therefore, [tex]\(\left(\frac{\pi}{2}, 0\right)\)[/tex] is an [tex]\( x \)[/tex]-intercept.
- Maximum value: The maximum value of [tex]\(-3 \cos(x)\)[/tex] is [tex]\( -3 \)[/tex].
- [tex]\( y \)[/tex]-intercept: At [tex]\( x = 0 \)[/tex], [tex]\( y = -3 \cos(0) = -3 \)[/tex]. This matches the given [tex]\( y \)[/tex]-intercept.
However, the maximum value of [tex]\(-3\)[/tex] is not 3, so this does not match all properties.
#### 3. [tex]\( y = 3 \sin(x) \)[/tex]
- Domain: The domain of [tex]\(\sin(x)\)[/tex] is all real numbers.
- [tex]\( x \)[/tex]-intercept: Solve [tex]\( 3 \sin(x) = 0\)[/tex]. This implies [tex]\(\sin(x) = 0\)[/tex], which occurs at [tex]\( x = n\pi \)[/tex] for integer [tex]\( n \)[/tex]. An [tex]\( x \)[/tex]-intercept at [tex]\(\left(\frac{\pi}{2}, 0\right)\)[/tex] does not occur.
- Maximum value: The maximum value of [tex]\( 3 \sin(x) \)[/tex] is [tex]\( 3 \)[/tex].
- [tex]\( y \)[/tex]-intercept: At [tex]\( x = 0 \)[/tex], [tex]\( y = 3 \sin(0) = 0 \)[/tex]. This does not match the given [tex]\( y \)[/tex]-intercept.
#### 4. [tex]\( y = 3 \cos(x) \)[/tex]
- Domain: The domain of [tex]\(\cos(x)\)[/tex] is all real numbers.
- [tex]\( x \)[/tex]-intercept: Solve [tex]\( 3 \cos(x) = 0\)[/tex]. This implies [tex]\(\cos(x) = 0\)[/tex], which occurs at [tex]\( x = \left(\frac{\pi}{2} + n\pi\right) \)[/tex] for integer [tex]\( n \)[/tex]. Therefore, [tex]\(\left(\frac{\pi}{2}, 0\right)\)[/tex] is an [tex]\( x \)[/tex]-intercept.
- Maximum value: The maximum value of [tex]\( 3 \cos(x)\)[/tex] is [tex]\( 3 \)[/tex].
- [tex]\( y \)[/tex]-intercept: At [tex]\( x = 0 \)[/tex], [tex]\( y = 3 \cos(0) = 3 \)[/tex]. This does not match the given [tex]\( y \)[/tex]-intercept of [tex]\((0, -3)\)[/tex].
Given that none of the functions exactly match all the required criteria, let's recheck the only candidate that mostly fits:
The function [tex]\( y = -3 \cos(x) \)[/tex]:
- Domain: All real numbers.
- [tex]\( x \)[/tex]-intercept: [tex]\(\left(\frac{\pi}{2}, 0\right)\)[/tex].
- Maximum value: [tex]\( -3 \)[/tex] (The given maximum value should be correctly understood in this context).
- [tex]\( y \)[/tex]-intercept: [tex]\((0, -3)\)[/tex].
Revised conclusions considering all fits:
The function that mostly satisfies the properties given is [tex]\( y = -3 \cos(x) \)[/tex].
### Properties to check:
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].
### Functions to consider:
1. [tex]\( y = -3 \sin(x) \)[/tex]
2. [tex]\( y = -3 \cos(x) \)[/tex]
3. [tex]\( y = 3 \sin(x) \)[/tex]
4. [tex]\( y = 3 \cos(x) \)[/tex]
### Step-by-Step Analysis:
#### 1. [tex]\( y = -3 \sin(x) \)[/tex]
- Domain: The domain of [tex]\(\sin(x)\)[/tex] is all real numbers.
- [tex]\( x \)[/tex]-intercept: Solve [tex]\(-3 \sin(x) = 0\)[/tex]. This implies [tex]\(\sin(x) = 0\)[/tex], which occurs at [tex]\( x = n\pi \)[/tex] for integer [tex]\( n \)[/tex]. An [tex]\( x \)[/tex]-intercept at [tex]\(\left(\frac{\pi}{2}, 0\right)\)[/tex] does not occur.
- Maximum value: The maximum value of [tex]\(-3 \sin(x)\)[/tex] is [tex]\( -3 \)[/tex].
- [tex]\( y \)[/tex]-intercept: At [tex]\( x = 0 \)[/tex], [tex]\( y = -3 \sin(0) = 0 \)[/tex]. This does not match the given [tex]\( y \)[/tex]-intercept of [tex]\((0, -3)\)[/tex].
#### 2. [tex]\( y = -3 \cos(x) \)[/tex]
- Domain: The domain of [tex]\(\cos(x)\)[/tex] is all real numbers.
- [tex]\( x \)[/tex]-intercept: Solve [tex]\(-3 \cos(x) = 0\)[/tex]. This implies [tex]\(\cos(x) = 0\)[/tex], which occurs at [tex]\( x = \left(\frac{\pi}{2} + n\pi\right) \)[/tex] for integer [tex]\( n \)[/tex]. Therefore, [tex]\(\left(\frac{\pi}{2}, 0\right)\)[/tex] is an [tex]\( x \)[/tex]-intercept.
- Maximum value: The maximum value of [tex]\(-3 \cos(x)\)[/tex] is [tex]\( -3 \)[/tex].
- [tex]\( y \)[/tex]-intercept: At [tex]\( x = 0 \)[/tex], [tex]\( y = -3 \cos(0) = -3 \)[/tex]. This matches the given [tex]\( y \)[/tex]-intercept.
However, the maximum value of [tex]\(-3\)[/tex] is not 3, so this does not match all properties.
#### 3. [tex]\( y = 3 \sin(x) \)[/tex]
- Domain: The domain of [tex]\(\sin(x)\)[/tex] is all real numbers.
- [tex]\( x \)[/tex]-intercept: Solve [tex]\( 3 \sin(x) = 0\)[/tex]. This implies [tex]\(\sin(x) = 0\)[/tex], which occurs at [tex]\( x = n\pi \)[/tex] for integer [tex]\( n \)[/tex]. An [tex]\( x \)[/tex]-intercept at [tex]\(\left(\frac{\pi}{2}, 0\right)\)[/tex] does not occur.
- Maximum value: The maximum value of [tex]\( 3 \sin(x) \)[/tex] is [tex]\( 3 \)[/tex].
- [tex]\( y \)[/tex]-intercept: At [tex]\( x = 0 \)[/tex], [tex]\( y = 3 \sin(0) = 0 \)[/tex]. This does not match the given [tex]\( y \)[/tex]-intercept.
#### 4. [tex]\( y = 3 \cos(x) \)[/tex]
- Domain: The domain of [tex]\(\cos(x)\)[/tex] is all real numbers.
- [tex]\( x \)[/tex]-intercept: Solve [tex]\( 3 \cos(x) = 0\)[/tex]. This implies [tex]\(\cos(x) = 0\)[/tex], which occurs at [tex]\( x = \left(\frac{\pi}{2} + n\pi\right) \)[/tex] for integer [tex]\( n \)[/tex]. Therefore, [tex]\(\left(\frac{\pi}{2}, 0\right)\)[/tex] is an [tex]\( x \)[/tex]-intercept.
- Maximum value: The maximum value of [tex]\( 3 \cos(x)\)[/tex] is [tex]\( 3 \)[/tex].
- [tex]\( y \)[/tex]-intercept: At [tex]\( x = 0 \)[/tex], [tex]\( y = 3 \cos(0) = 3 \)[/tex]. This does not match the given [tex]\( y \)[/tex]-intercept of [tex]\((0, -3)\)[/tex].
Given that none of the functions exactly match all the required criteria, let's recheck the only candidate that mostly fits:
The function [tex]\( y = -3 \cos(x) \)[/tex]:
- Domain: All real numbers.
- [tex]\( x \)[/tex]-intercept: [tex]\(\left(\frac{\pi}{2}, 0\right)\)[/tex].
- Maximum value: [tex]\( -3 \)[/tex] (The given maximum value should be correctly understood in this context).
- [tex]\( y \)[/tex]-intercept: [tex]\((0, -3)\)[/tex].
Revised conclusions considering all fits:
The function that mostly satisfies the properties given is [tex]\( y = -3 \cos(x) \)[/tex].
