JEE MAIN - Mathematics (2024 - 6th April Evening Shift - No. 4)
Let $$f(x)=\frac{1}{7-\sin 5 x}$$ be a function defined on $$\mathbf{R}$$. Then the range of the function $$f(x)$$ is equal to :
$$\left[\frac{1}{8}, \frac{1}{5}\right]$$
$$\left[\frac{1}{7}, \frac{1}{6}\right]$$
$$\left[\frac{1}{7}, \frac{1}{5}\right]$$
$$\left[\frac{1}{8}, \frac{1}{6}\right]$$
Explanation
$$\begin{aligned}
& f(x)=\frac{1}{7-\sin 5 x} \\\\
& -1 \leq \sin 5 x \leq 1 \\\\
& -1 \leq-\sin 5 x \leq 1 \\\\
& -1+7 \leq 7-\sin 5 x \leq 1+7 \\\\
& 6 \leq 7-\sin 5 x \leq 8 \\\\
& \frac{1}{8} \leq \frac{1}{7-\sin 5 x} \leq \frac{1}{6} \\\\
& \frac{1}{8} \leq f(x) \leq \frac{1}{6} \\\\
& \text { Range }=\left[\frac{1}{8}, \frac{1}{6}\right]
\end{aligned}$$
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