Consider the function y=sqrt(1+cos2x)/2
(everything is under the square root)
a. Can x take on any real value?
b. How large can cos 2x become? How small?
c. How large can 1+cos2x/2 become? How small?
d. What are the domain and range of the original function?
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For A, I put that x can be any real value, but the entire function would still be an irrational number.
As for B and C, I'm not exactly sure what values it's asking for. Like, how large or small can the values be without becoming a nonreal number?
And I think I could figure out D if I can figure out B and C. Help?
Thanks!
b. How large can cos 2x become? How small?
You need to supply the range of cos(2x).
c. How large can 1+cos2x/2 become? How small?
Similarly, you need to supply the range of (1+cos(2x))/2
These two questions help you answer #D, as you corrected pointed out.
a. Yes, x can take on any real value. The function is defined for all real values of x.
b. To determine the largest and smallest possible values of cos 2x, we need to consider the range of the cosine function. The cosine function oscillates between -1 and 1. Therefore, cos 2x can take on values between -1 and 1. The largest possible value for cos 2x is 1, which occurs when 2x is a multiple of 2π (e.g., 0, 2π, 4π, etc.). The smallest possible value for cos 2x is -1, which occurs when 2x is a multiple of π (e.g., π, 3π, 5π, etc.).
c. To determine how large or small 1 + cos 2x / 2 can become, let's consider the range of the cosine function again. As discussed in part b, cos 2x can take on values between -1 and 1. Dividing by 2, we now have a range of -1/2 to 1/2. Adding 1 to this range gives us a range of 1/2 to 3/2. Therefore, 1 + cos 2x / 2 can take on values between 1/2 and 3/2.
d. The domain of the original function y = sqrt(1 + cos 2x)/2 is all real numbers because x can take on any real value. However, the range of the function is restricted by the range of cos 2x, as discussed in part b. Therefore, the range of the function is [1/2, 3/2].
Let's go through each question step by step:
a. Can x take on any real value?
In the given function, there are no restrictions on the input variable x. Therefore, x can indeed take on any real value.
b. How large can cos 2x become? How small?
The range of the cosine function is -1 to 1. When we have cos 2x, it means taking the cosine of twice the value of x. Since cosine repeats its values every 2π radians (360 degrees), we can say that cos 2x can take on any value between -1 and 1, inclusive. So, -1 ≤ cos 2x ≤ 1.
c. How large can 1+cos2x/2 become? How small?
To find the maximum and minimum values of the expression 1 + cos 2x/2, we need to analyze the effect of the cosine function on its range. As previously mentioned, cos 2x can range from -1 to 1.
If we divide cos 2x by 2, the range of the resulting expression becomes -0.5 to 0.5, because (-1/2) ≤ cos 2x/2 ≤ (1/2).
Finally, if we add 1 to cos 2x/2, the range shifts upwards by 1 unit. This means that the maximum value of 1 + cos 2x/2 is 1 + (1/2) = 3/2, and the minimum value is 1 - (1/2) = 1/2.
So, the expression 1 + cos 2x/2 can range from 1/2 to 3/2.
d. What are the domain and range of the original function?
The domain of the original function is the set of all real numbers, which we established in part a. Therefore, there are no restrictions on the x-values we can use.
The range of the original function is given by the range of the expression 1 + cos 2x/2, which we determined to be 1/2 to 3/2. So, the range of the original function y = sqrt(1 + cos 2x)/2 is [1/2, 3/2].
I hope this clarifies the solutions for all the questions. Let me know if you have any further queries!