Question

find the range of 1/5- cos 3x

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Answer 1

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm :\longmapsto\:\dfrac{1}{5 - cos3x}$

Let assume that

$\rm :\longmapsto\:f(x) = \dfrac{1}{5 - cos3x}$

To find the range, Let first recall the definition of range.

Range of a function f(x) is defined as set of result of those values which are assumed by x.

Now, we know

$\rm :\longmapsto\: - 1 \leqslant cos3x \leqslant 1$

On multiply each term by - 1, we get

$\rm :\longmapsto\:1 \geqslant - cos3x \geqslant - 1$

can be rewritten as

$\rm :\longmapsto\: - 1 \leqslant - cos3x \leqslant 1$

On adding 5 in each term, we get

$\rm :\longmapsto\:5 - 1 \leqslant 5 - cos3x \leqslant 5 + 1$

$\rm :\longmapsto\:4 \leqslant 5 - cos3x \leqslant 6$

$\rm :\longmapsto\:\dfrac{1}{6} \leqslant \dfrac{1}{5 - cos3x} \leqslant \dfrac{1}{4}$

$\rm :\longmapsto\:\dfrac{1}{6} \leqslant f(x) \leqslant \dfrac{1}{4}$

$\bf\implies \:f(x) \: \in \: \bigg[\dfrac{1}{6}, \: \dfrac{1}{4} \bigg]$

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Learn More :-

Domain of a function f(x) is defined as set of those values of x for which function is well defined or for which function assumes real values.