Question

The range of the function $f\left( x \right) = \dfrac{1}{{3 + \sin x}}$ is
A. $\left[ { - 1,1} \right]$
B. $\left[ {\dfrac{1}{4},\dfrac{1}{2}} \right]$

Answer 1

Hint : In this question, we need to find the range, i.e. the maximum and minimum value of the given function $f\left( x \right)$. Here, the range of the function only depends upon the range of the sine function
because 3 is a constant. So, we are going to use the range of the sine function, add 3 to it and then use the operation of reciprocal, all in all, transform the sine function into the one given in the question. And while performing all the operations on the sine function, we are also going to always perform the same operations simultaneously on the range of the function.
Formula Used:
We are going to use the formula of the range of the sine function, which is:
$- 1 \le \sin x \le 1$

Complete step-by-step answer :
For solving this, we are going to write down the sine function with its range. Then we are going to apply some operations on it to transform $\sin x$ into $\dfrac{1}{{3 + \sin x}}$ and at the same time, apply the same operations on the range. When the $\sin x$ reaches the required form, the transformed range is going to give us the answer.
So, we have,
$- 1 \le \sin x \le 1$
Adding 3,
$2 \le 3 + \sin x \le 4$
Taking reciprocal (taking reciprocal inverts the inequality)
$\dfrac{1}{2} \ge \dfrac{1}{{3 + \sin x}} \ge \dfrac{1}{4}$
or, $\dfrac{1}{4} \le \dfrac{1}{{3 + \sin x}} \le \dfrac{1}{2}$
Since the answer does not contain a zero (because zero cannot be in the range as any non-zero number divided by any other non-zero number can never give a zero), we have the answer containing all the values in the acceptable range.
So, the correct answer is “Option B”.

Note : So, we saw that in solving questions of this type, we do not need any complex or complicated equations or formulae. We just need to write down the range of the trigonometric function, then add, subtract, multiply or divide or apply whatever operations are needed to be applied and transform the crude, basic, raw function into the one given in the question. And while applying the operations on the function, we need to apply the same set of exact operations, in the same order on the range of the function. And when the function gets transformed into the one given in the question, the transformed range is going to give us the answer.