Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Experience the ease of finding accurate answers to your questions from a knowledgeable community of professionals. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
Let's analyze the function [tex]\( f(x) = 5 \cos(x) + 1 \)[/tex] step by step.
### Period
The period of a standard cosine function [tex]\( \cos(x) \)[/tex] is [tex]\( 2\pi \)[/tex]. Shifting and scaling the function vertically or horizontally does not affect the period. Therefore, the period of [tex]\( 5 \cos(x) + 1 \)[/tex] is also [tex]\( 2\pi \)[/tex], not [tex]\( 10\pi \)[/tex].
So, the statement:
- "The period of the function is [tex]\( 10\pi \)[/tex]."
is False.
### Amplitude
The amplitude of a cosine function [tex]\( a \cos(x) \)[/tex] is given by the coefficient [tex]\( a \)[/tex]. In this case, the coefficient is 5. Thus, the amplitude of [tex]\( f(x) \)[/tex] is 5, not 2.5.
So, the statement:
- "The function has an amplitude of 2.5."
is False.
### Zeroes of the Function
A zero of the function [tex]\( f(x) \)[/tex] is a value [tex]\( x \)[/tex] such that [tex]\( f(x) = 0 \)[/tex].
Let's find whether [tex]\( f \left( \frac{\pi}{2} \right) = 0 \)[/tex]:
[tex]\[ f \left( \frac{\pi}{2} \right) = 5 \cos \left( \frac{\pi}{2} \right) + 1 = 5 \cdot 0 + 1 = 1. \][/tex]
Since [tex]\( f \left( \frac{\pi}{2} \right) \neq 0 \)[/tex], [tex]\(\left( \frac{\pi}{2}, 0 \right)\)[/tex] is not a zero of the function.
So, the statement:
- "A zero of the function is [tex]\( \left( \frac{\pi}{2}, 0 \right) \)[/tex]."
is False.
### Range
The range of a cosine function [tex]\( a \cos(x) + b \)[/tex] is determined by the amplitude [tex]\( a \)[/tex] and the vertical shift [tex]\( b \)[/tex]. Since [tex]\(\cos(x)\)[/tex] oscillates between -1 and 1, we have:
[tex]\[ -1 \leq \cos(x) \leq 1. \][/tex]
Multiplying by 5:
[tex]\[ -5 \leq 5 \cos(x) \leq 5. \][/tex]
Adding 1:
[tex]\[ -4 \leq 5 \cos(x) + 1 \leq 6. \][/tex]
Thus, the range of [tex]\( f(x) = 5 \cos(x) + 1 \)[/tex] is indeed [tex]\( -4 \leq y \leq 6 \)[/tex].
So, the statement:
- "The range of the function is the set of real numbers [tex]\( -4 \leq y \leq 6 \)[/tex]."
is True.
In conclusion:
- The period of the function is [tex]\( 10\pi \)[/tex]. False.
- The function has an amplitude of 2.5. False.
- A zero of the function is [tex]\( \left( \frac{\pi}{2}, 0 \right) \)[/tex]. False.
- The range of the function is [tex]\( -4 \leq y \leq 6 \)[/tex]. True.
### Period
The period of a standard cosine function [tex]\( \cos(x) \)[/tex] is [tex]\( 2\pi \)[/tex]. Shifting and scaling the function vertically or horizontally does not affect the period. Therefore, the period of [tex]\( 5 \cos(x) + 1 \)[/tex] is also [tex]\( 2\pi \)[/tex], not [tex]\( 10\pi \)[/tex].
So, the statement:
- "The period of the function is [tex]\( 10\pi \)[/tex]."
is False.
### Amplitude
The amplitude of a cosine function [tex]\( a \cos(x) \)[/tex] is given by the coefficient [tex]\( a \)[/tex]. In this case, the coefficient is 5. Thus, the amplitude of [tex]\( f(x) \)[/tex] is 5, not 2.5.
So, the statement:
- "The function has an amplitude of 2.5."
is False.
### Zeroes of the Function
A zero of the function [tex]\( f(x) \)[/tex] is a value [tex]\( x \)[/tex] such that [tex]\( f(x) = 0 \)[/tex].
Let's find whether [tex]\( f \left( \frac{\pi}{2} \right) = 0 \)[/tex]:
[tex]\[ f \left( \frac{\pi}{2} \right) = 5 \cos \left( \frac{\pi}{2} \right) + 1 = 5 \cdot 0 + 1 = 1. \][/tex]
Since [tex]\( f \left( \frac{\pi}{2} \right) \neq 0 \)[/tex], [tex]\(\left( \frac{\pi}{2}, 0 \right)\)[/tex] is not a zero of the function.
So, the statement:
- "A zero of the function is [tex]\( \left( \frac{\pi}{2}, 0 \right) \)[/tex]."
is False.
### Range
The range of a cosine function [tex]\( a \cos(x) + b \)[/tex] is determined by the amplitude [tex]\( a \)[/tex] and the vertical shift [tex]\( b \)[/tex]. Since [tex]\(\cos(x)\)[/tex] oscillates between -1 and 1, we have:
[tex]\[ -1 \leq \cos(x) \leq 1. \][/tex]
Multiplying by 5:
[tex]\[ -5 \leq 5 \cos(x) \leq 5. \][/tex]
Adding 1:
[tex]\[ -4 \leq 5 \cos(x) + 1 \leq 6. \][/tex]
Thus, the range of [tex]\( f(x) = 5 \cos(x) + 1 \)[/tex] is indeed [tex]\( -4 \leq y \leq 6 \)[/tex].
So, the statement:
- "The range of the function is the set of real numbers [tex]\( -4 \leq y \leq 6 \)[/tex]."
is True.
In conclusion:
- The period of the function is [tex]\( 10\pi \)[/tex]. False.
- The function has an amplitude of 2.5. False.
- A zero of the function is [tex]\( \left( \frac{\pi}{2}, 0 \right) \)[/tex]. False.
- The range of the function is [tex]\( -4 \leq y \leq 6 \)[/tex]. True.
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.