Question

Which of the functions have a range of all real numbers greater than or equal to 1 or less than or equal to -17 Check all that apply.

Answer 1

Hello there. To solve this question, we'll have to remember some properties about trigonometric functions.

We want to determine which of the following functions has a range of all real numbers greater than or equal to 1 or less than or equal to -1.

a) y = sec x

b) y = csc x

c) y = tan x

d) y = cot x

First, remember some properties about each of the functions:

sec x can be expressed in terms of cosine, as follows:

`\sec(x)=(1)/(\cos(x))`

And of course, it doesn't exists for values of x that are odd multiples of pi/2, we say

`\sec(x)\text{ DNE for }x=((2k-1)\pi)/(2),\text{ }k=\mathbb{Z}`

Those are the points for which cosine is zero. We also know that

`-1<\cos(x)<1`

So the only thing we're making here is in fact eliminating the zero from the interval, making it an union:

`[-1,0)\cup(0,1]`

So the secant can have values that are greater than or equal to 1 or less than or equal to -1 in that interval. We write

`\{y\in\mathbb{R}\,\,\,|\,\,\,y\leq-1\,\text{or}\,y\geq1\}`

For tan x, we can also express it in terms of sines and cosines:

`\tan(x)=(\sin(x))/(\cos(x))`

So it happens to have the same property as secant, since there is a cosine in the denominator, we cannot have odd multiples of pi/2, hence

`\tan(x)\text{ DNE for }x=((2k-1)\pi)/(2),\,k=\mathbb{Z}`

Again, we're eliminating the zero from the range of cosine and hence tan(x) has a range of all real numbers (assintotically going towards infinity).

For csc x, we have something different. It is defined as:

`\csc(x)=(1)/(\sin(x))`

Now it does not exists for integer multiples of pi, hence we say

`\csc(x)\text{ DNE for }x=k\pi,\,k\in\mathbb{Z}`

And as

`-1<\sin(x)<1`

We're taking out the zeros from this interval, therefore it has the same behavior as the secant and their range is also the same: All real numbers greater than or equal to 1 or less than or equal to -1.

Finally, we have the function cot x.

Since it is defined as:

`\cot(x)=(1)/(\tan(x))=(\cos(x))/(\sin(x))`

We're eliminating its zeros for odd multiples of pi/2 when rewriting it into the second expression.

But it also goes assintotically towards infinity when getting closer to these points, which means that the range of this function is all real numbers.

The answer to this question are the only functions that applies: sec x and csc x.