Question

The domain and range of \[y\text{ }=\text{ }f\left( x \right)\text{ }=\text{ }tan\left( sin\,x\text{ }+\text{ }cos\text{ }x \right)\] are, choose the correct option:
A.\[D(f)=R,\,R(f)=\left[ -\tan \sqrt{2},\tan \sqrt{2} \right]\]
B.\[D(f)=R,\,R(f)=\left[ 0,\tan \sqrt{2} \right]\]
C.\[D(f)=R,\,R(f)=\left( -\tan \sqrt{2},\tan \sqrt{2} \right)\]
D.\[D(f)=R,\,R(f)=\left( 0,\tan \sqrt{2} \right)\]

Answer 1

Hint: In order to find the domain, we have to find all those points where the function \[y\text{ }=\text{ }f\left( x \right)\text{ }=\text{ }tan\left( sin\,x\text{ }+\text{ }cos\text{ }x \right)\]is defined. Next, in order to find the range of the given function, we have to find the maximum and the minimum value and thus all the values inside these two extremes will be the range.

Complete step-by-step answer:
In the question, we have to find the domain and range of the function\[y\text{ }=\text{ }f\left( x \right)\text{ }=\text{ }tan\left( sin\,x\text{ }+\text{ }cos\text{ }x \right)\]. So, first we will find the domain, which is set of all the points where the function is defined. In other words, we can find the points where the function is not defined and then all remaining points in the real number set will be defined. Now, here we know that sin x and cos x is defined for all real numbers (R). So, the tangent of \[\left( sin\,x\text{ }+\text{ }cos\text{ }x \right)\]will also be defined for all real numbers, as there is no point where the function is not defined.
So, finally, we can say the domain of \[y\text{ }=\text{ }f\left( x \right)\text{ }=\text{ }tan\left( sin\,x\text{ }+\text{ }cos\text{ }x \right)\] is all real numbers and is written as: \[D(f)=R\].
Next, we will find the range of the function \[y\text{ }=\text{ }f\left( x \right)\text{ }=\text{ }tan\left( sin\,x\text{ }+\text{ }cos\text{ }x \right)\]. So, at first we will find the maximum value of \[\left( sin\,x\text{ }+\text{ }cos\text{ }x \right)\], then the minimum value of \[\left( sin\,x\text{ }+\text{ }cos\text{ }x \right)\], because all the values lying in between two extremes will be the range of the function. Also, since there is no point where the function is not defined.
So, to find the maximum value of \[\left( sin\,x\text{ }+\text{ }cos\text{ }x \right)\], we will use the concept that sin x in increasing when cos x is decreasing, so at 45 degrees both are same i.e., \[\sin x=\cos x=\dfrac{1}{\sqrt{2}}\]. Hence the maximum value of \[\left( sin\,x\text{ }+\text{ }cos\text{ }x \right)=\dfrac{2}{\sqrt{2}}=\sqrt{2}\]. Similarly, the minimum value is when both are equal to \[\sin x=\cos x=-\dfrac{1}{\sqrt{2}}\], so the minimum value of \[\left( sin\,x\text{ }+\text{ }cos\text{ }x \right)=-\dfrac{2}{\sqrt{2}}=-\sqrt{2}\].
Now, the maximum value of \[y\text{ }=\text{ }f\left( x \right)\text{ }=\text{ }tan\left( sin\,x\text{ }+\text{ }cos\text{ }x \right)=tan\left( \sqrt{2} \right)\] and the minimum value is \[y\text{ }=\text{ }f\left( x \right)\text{ }=\text{ }tan\left( sin\,x\text{ }+\text{ }cos\text{ }x \right)=-tan\left( \sqrt{2} \right)\]. So, the range is all the numbers in between these two extreme values including end points, and is shown as follows: \[R(f)=\left[ -\tan \sqrt{2},\tan \sqrt{2} \right]\]
So finally, the domain and range is given as; \[D(f)=R,\,R(f)=\left[ -\tan \sqrt{2},\tan \sqrt{2} \right]\]. Hence the correct answer is option A.

Note: It can be noted that when the function has the value infinity, then we don’t say that it is not defined at that point, but we say that it is discontinuous at that point. So here, when we are finding range we will take care of such points.