Question

If \( f : \mathbb{R} \to A \), defined by \( f(x) = \cos x + \sqrt{3}\sin x - 1 \), is an onto function, then \( A = \)

• A. \([-1, 2]\)
• B. \([- \sqrt{3}, \sqrt{3}]\)
• C. \([-3, 1]\)
• D. \([-2, 2]\)

Hint:

When given a trigonometric expression in the form \( a\cos x + b\sin x \), rewrite it using amplitude-phase form: \( R\sin(x+\alpha) \).

Answer 1

The Correct Option is C

Let \( f(x) = \cos x + \sqrt{3} \sin x - 1 \). This is a linear combination of sine and cosine: \[ f(x) = R \sin(x + \alpha) - 1 \] where \( R = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{4} = 2 \). So, maximum and minimum of \( f(x) \) are: \[ \max f(x) = 2 - 1 = 1,
\min f(x) = -2 - 1 = -3 \] Thus, the range of \( f(x) \) is \( [-3, 1] \). Since \( f \) is onto, \( A = [-3, 1] \).