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The range of $f\left( x \right)=\cos \left( \dfrac{x}{3} \right)$ is:
A. $\left( -\dfrac{1}{3},\dfrac{1}{3} \right)$
B. $\left[ -1,1 \right]$
C. $\left( \dfrac{1}{3},-\dfrac{1}{3} \right)$
D. $\left( -3,3 \right)$

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Last updated date: 13th Jun 2024
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Answer
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Hint: Here, we have been asked to give the range of the function defined as $f\left( x \right)=\cos \left( \dfrac{x}{3} \right)$. To do this, we will first have to define what kind of function this is so that we get its better understanding. Here, it is a trigonometric function. Then we will give its domain and from all the collective knowledge that we have of the function now, we will provide its range.

Complete step by step answer:
We have been given a function $f\left( x \right)=\cos \left( \dfrac{x}{3} \right)$ and we need to find its range. For this we first need to know what kind of function this is.
This is a trigonometric function and we know that the domain of all trigonometric functions (not inverse, only the simple trigonometric functions) is all real numbers.
Thus, here also in f(x), all the real numbers can take the value of x.
But we also know that the cosine function is a periodic function, i.e. the values of cosx start repeating after a fixed interval.
Now, we also know that the value of cosx always lies in the interval $\left[ -1,1 \right]$ no matter whatever the value of x be.
Here, we have been given $\cos \left( \dfrac{x}{3} \right)$ and we know that $\dfrac{x}{3}$ can also take any real value. Hence, the range of $\cos \left( \dfrac{x}{3} \right)$ is the same as that of cosx.
Hence, the range of $f\left( x \right)=\cos \left( \dfrac{x}{3} \right)$ is [-1,1].

So, the correct answer is “Option B”.

Note: Here we have given ranges of some trigonometric functions which might come in handy:
1. sinx: [-1,1]
2. cosx: [-1,1]
3. tanx: $\left( -\infty ,\infty \right)$
4. cotx: $\left( -\infty .\infty \right)$
5. secx: $(-\infty ,-1]\cup [1,\infty )$
6. cosecx: $(-\infty ,-1]\cup [1,\infty )$