Sets, Relations & Functions
Range of Functions / Logarithmic Functions
nta_pyq_2023_jan
Grade 11
Question:
Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \log_{\sqrt{m}}\left\{\sqrt{2}(\sin x - \cos x) + m - 2\right\}$, for some $m$, such that the range of $f$ is $[0, 2]$. Then the value of $m$ is
Step-by-Step Solution
Key Concept: Range of $\sqrt{2}(\sin x - \cos x)$ is $[-2,2]$; match with log conditions to find $m$.
Let $k=\sqrt{2}(\sin x-\cos x)\in[-2,2]$. Range of $f$ is $[0,2]$ requires $1 \le k+m-2 \le m$, giving $-m+3 \le k \le 2$. Equating: $-m+3=-2 \Rightarrow m=5$.
Correct Answer: 1