Find the range of $f(x) = \frac{1}{2\cos

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Target Exam

CUET

Subject

-- Mathematics - Section B1

Chapter

Relations and Functions

Question:

Find the range of $f(x) = \frac{1}{2\cos x−1}$.

Options:

$(-∞,-\frac{1}{3}]∪[-1,-∞)$

$(-∞,-\frac{1}{3}]∪[1,∞)$

$(∞,\frac{1}{3}]∪[-1,-∞)$

$(-∞,\frac{1}{3}]∪[-1,0)$

Correct Answer:

$(-∞,-\frac{1}{3}]∪[1,∞)$

Explanation:

$-1≤ \cos x ≤ 1$

or $-2 ≤ 2 \cos x ≤2$

or $-3≤2 \cos x-1≤1$

For $\frac{1}{2\cos x−1}$,

$-3≤2 \cos x-1<0$ or $0<2\cos x - 1 ≤1$

i.e., $-∞<\frac{1}{2\cos x-1}≤\frac{-1}{3}$ or $1≤\frac{1}{2\cos x- 1}<∞$

Hence, the range is $(-∞,-\frac{1}{3}]∪[1,∞)$