Sure, let's go through each part of the question step by step.
### (i) Domain of [tex]\( f(x) = \frac{1}{\sqrt{1 + \cos(x)}} \)[/tex]
To determine the domain of this function, we need to ensure the denominator is not zero and the expression under the square root is non-negative:
- The denominator [tex]\( \sqrt{1 + \cos(x)} \)[/tex] must be non-zero. This means [tex]\( 1 + \cos(x) \neq 0 \)[/tex].
- [tex]\( 1 + \cos(x) \)[/tex] is non-zero when [tex]\( \cos(x) \neq -1 \)[/tex].
The cosine function [tex]\(\cos(x)\)[/tex] equals [tex]\(-1\)[/tex] at [tex]\( x = (2n+1)\pi \)[/tex] where [tex]\( n \)[/tex] is an integer.
Therefore, the domain of the function [tex]\( \frac{1}{\sqrt{1+\cos(x)}} \)[/tex] is [tex]\( R - (2n+1)\pi \)[/tex].
Answer: (b) [tex]\( R - (2n+1)\pi \)[/tex]
### (ii) Domain of [tex]\( f(x) = \frac{1}{\sin(x) + \cos(x)} \)[/tex]
For the function to be defined, the denominator [tex]\(\sin(x) + \cos(x)\)[/tex] must not be zero. We need to find when [tex]\(\sin(x) + \cos(x) = 0\)[/tex]:
- This equation holds true when [tex]\( \sin(x) = -\cos(x) \)[/tex].
The expression [tex]\(\sin(x) = -\cos(x)\)[/tex] is true at [tex]\( x = (4n+1)\frac{\pi}{4} \)[/tex] where [tex]\( n \)[/tex] is an integer.
Therefore, the domain of the function [tex]\( \frac{1}{\sin(x) + \cos(x)} \)[/tex] is [tex]\( R - (4n+1)\frac{\pi}{4} \)[/tex].
Answer: (b) [tex]\( R - (4n+1)\frac{\pi}{4} \)[/tex]
### (iii) Range of [tex]\( f(x) = \sqrt{5 + 4\cos(x)} \)[/tex]
To find the range of this function, we need to look at the expression inside the square root, which is [tex]\( 5 + 4\cos(x) \)[/tex]:
- The cosine function [tex]\(\cos(x)\)[/tex] has a range of [tex]\([-1, 1]\)[/tex].
- Therefore, [tex]\( 4\cos(x) \)[/tex] ranges from [tex]\(-4\)[/tex] to [tex]\(4\)[/tex].
- Adding 5 to this range, we get [tex]\( 5 + 4\cos(x) \)[/tex], which ranges from [tex]\( 5 - 4 = 1 \)[/tex] to [tex]\( 5 + 4 = 9 \)[/tex].
- Taking the square root of these values, we get the range of [tex]\( \sqrt{5 + 4\cos(x)} \)[/tex] as [tex]\([ \sqrt{1}, \sqrt{9} ] = [1, 3]\)[/tex].
Answer: The range of [tex]\( f(x) = \sqrt{5 + 4\cos(x)} \)[/tex] is (1, 3)
