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Functions question

2024 · 6 Apr · Shift 2 · Q33

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Functions
2024 · 6 Apr · Shift 2 · Q33

Functions question

2024 · 6 Apr · Shift 2 · Q33

JEE MainMathematicsFunctionsMCQ+4 / −1

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 :

A
$\left[\frac{1}{8}, \frac{1}{5}\right]$
B
$\left[\frac{1}{7}, \frac{1}{6}\right]$
C
$\left[\frac{1}{7}, \frac{1}{5}\right]$
D
$\left[\frac{1}{8}, \frac{1}{6}\right]$

View written solutionFree

Correct answer: D

We are given $f(x)=\frac{1}{7-\sin 5x}.$ We need the range of $f(x)$ for all $x\in \mathbb{R}$ .
Since $-1\le \sin 5x\le 1,$ we get the range of the denominator: $7-1\le 7-\sin 5x\le 7-(-1).$ So, $6\le 7-\sin 5x\le 8.$
Therefore, $f(x)=\frac{1}{7-\sin 5x}$ will take values as the reciprocal of numbers in $[6,8]$ . Since $\frac{1}{t}$ is decreasing for $t>0$ , the interval reverses: $\frac{1}{8}\le f(x)\le \frac{1}{6}.$ Thus the range is $\left[\frac{1}{8},\frac{1}{6}\right].$
Check options:

A: $\left[\frac{1}{8},\frac{1}{5}\right]$ ❌
B: $\left[\frac{1}{7},\frac{1}{6}\right]$ ❌
C: $\left[\frac{1}{7},\frac{1}{5}\right]$ ❌
D: $\left[\frac{1}{8},\frac{1}{6}\right]$ ✅

Hence, the correct option is D.

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