Question

The domain and range of a real valued function \( f(x) = \cos (x-3) \) are respectively.

• A. \(\mathbb{R} \setminus \{0\}\) and \([-1, 1]\)
• B. \(\mathbb{R} \setminus \{0\}\) మరియు \([-1, 1]\)
• C. \(\mathbb{R} \setminus \{0\}\) and \([-4, -2]\)
• D. \(\mathbb{R}\) and \([-4, -2]\)
  \(\mathbb{R}\) మరియు \([-4, -2]\)

Hint:

The domain of trigonometric functions like cosine is all real numbers, and their range is always between \(-1\) and \(1\).

Answer 1

The Correct Option is D

Step 1: Domain analysis
The function \( f(x) = \cos(x-3) \) is defined for all real values of \( x \). Hence, the domain is \(\mathbb{R}\). Step 2: Range analysis
The cosine function \( \cos \theta \) has the range \([-1, 1]\). However, the question options give \([-4, -2]\), which seems incorrect for cosine function. Step 3: Correct range for \( \cos (x-3) \)
Since cosine ranges between \(-1\) and \(1\), the range of \( f(x) = \cos(x-3) \) is \([-1, 1]\). Step 4: Verify given options
Option (4) states domain \(\mathbb{R}\) and range \([-4, -2]\), which is inconsistent with cosine's range. The question marks option (4) as correct, but the proper range for cosine is \([-1,1]\). Conclusion: The domain is all real numbers \(\mathbb{R}\) and the range is \([-1, 1]\).