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Which of the following is true for [tex]f(x) = 5 \cos(x) + 1[/tex]?

A. The period of the function is [tex]10 \pi[/tex].
B. The function has an amplitude of 2.5.
C. A zero of the function is [tex]\left(\frac{\pi}{2}, 0\right)[/tex].
D. The range of the function is the set of real numbers [tex]-4 \leq y \leq 6[/tex].

Sagot :

GinnyAnswer
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.
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