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What is the range of the function #y = csc x#?

Answer 1

Lily Anderson

The range of #cscx# is #(-infty,-1]cup[1,infty)#.

Let us look at some details.

We know

#|sinx| le 1#

by taking the reciprocal,

#Rightarrow |cscx|=1/|sinx| ge 1#

(Note that we need to switch the direction of inequality.)

Hence, its range is #(-infty,-1]cup[1,infty)#.

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Answer 2

Serenity Jones

The range of the function ( y = \csc(x) ) is the set of all possible values that the function can output. Since the cosecant function is the reciprocal of the sine function, its range will be the set of all real numbers except for values where the sine function is equal to zero, because division by zero is undefined.

The range of ( \sin(x) ) is from -1 to 1, inclusive. So, the values of ( \csc(x) ) will range from negative infinity to negative one, unioned with positive one to positive infinity, excluding where ( \sin(x) ) equals zero.

Therefore, the range of ( y = \csc(x) ) is ( (-\infty, -1] \cup [1, \infty) ), excluding the points where ( \sin(x) = 0 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.