Let's analyze each function step by step to determine their respective ranges:
Part (iii): Function [tex]\( f(x) = \sqrt{5 + 4 \cos x} \)[/tex]
1. The function inside the square root is [tex]\( 5 + 4 \cos x \)[/tex].
2. The cosine function, [tex]\( \cos x \)[/tex], ranges from -1 to 1.
3. Substituting these values into [tex]\( 5 + 4 \cos x \)[/tex]:
- When [tex]\( \cos x = -1 \)[/tex]: [tex]\( 5 + 4 \cdot (-1) = 1 \)[/tex].
- When [tex]\( \cos x = 1 \)[/tex]: [tex]\( 5 + 4 \cdot 1 = 9 \)[/tex].
4. Therefore, the expression inside the square root, [tex]\( 5 + 4 \cos x \)[/tex], ranges from 1 to 9.
5. Taking the square root of these values, we get:
- [tex]\( \sqrt{1} = 1 \)[/tex]
- [tex]\( \sqrt{9} = 3 \)[/tex]
6. Thus, the range of the function [tex]\( f(x) = \sqrt{5 + 4 \cos x} \)[/tex] is [tex]\([1, 3]\)[/tex].
Answer: (c) [tex]\([1, 3]\)[/tex]
Part (iv): Function [tex]\( f(x) = [\cos x] \)[/tex], where [ ] denotes greatest integer function
1. The cosine function, [tex]\( \cos x \)[/tex], ranges from -1 to 1.
2. The greatest integer function (floor function) applied to [tex]\( \cos x \)[/tex] will produce:
- For [tex]\(\cos x\)[/tex] in [tex]\([-1, -0.01]\)[/tex], the greatest integer is [tex]\(-1\)[/tex].
- For [tex]\(\cos x\)[/tex] in [tex]\([0, 0.99]\)[/tex], the greatest integer is [tex]\(0\)[/tex].
- For [tex]\(\cos x = 1\)[/tex], the greatest integer is [tex]\(1\)[/tex].
3. Therefore, the set of values taken by [tex]\( [\cos x] \)[/tex] is \{-1, 0, 1\}.
Answer: (d) \{-1, 0, 1\}
Part (v): Function [tex]\( f(x) = 4 \cos^2 x - 3 \sin^2 x + 5 \)[/tex]
1. Using the trigonometric identity [tex]\(\cos^2 x + \sin^2 x = 1\)[/tex], we can rewrite the function:
[tex]\[
f(x) = 4 \cos^2 x - 3 (1 - \cos^2 x) + 5 = 4 \cos^2 x - 3 + 3 \cos^2 x + 5 = 7 \cos^2 x + 2
\][/tex]
2. The range of [tex]\( \cos^2 x \)[/tex] is [tex]\([0, 1]\)[/tex] since the cosine function squared ranges from 0 to 1.
3. Substituting these values into [tex]\( 7 \cos^2 x + 2 \)[/tex]:
- When [tex]\( \cos^2 x = 0 \)[/tex]: [tex]\( 7 \cdot 0 + 2 = 2 \)[/tex].
- When [tex]\( \cos^2 x = 1 \)[/tex]: [tex]\( 7 \cdot 1 + 2 = 9 \)[/tex].
4. Thus, the range of the function [tex]\( f(x) = 4 \cos^2 x - 3 \sin^2 x + 5 \)[/tex] is [tex]\([2, 9]\)[/tex].
Answer: (a) [tex]\([2, 9]\)[/tex]
