Question

List the domain and range for the secant and cotangent functions.

Answer 1

domain is read from left to right ( x values )

range lowest y value to highest y value

Answer 2

domain is the set of x-values included in the function. Range is as stated above; the range is the value of the output for the members of the domain.

Answer 3

set of values*

Answer 4

It's easier to work off a graph.

The secant function is a reciprocal of the cosine function
Hence the maximum values of the cosine function when inverted become minimum values, and zeroes of the cosine function become vertical asymptotes (1/0 or -1/0 -> +/-infinity).


So the domain of secant includes all real x apart from where these vertical asymptotes occur. The zeros of cos(x) are at x= pi/2, 3pi/2, etc. (you can see from your table) or in general
\[x = (2n+1) \frac{ \pi }{ 2 }\] for some integer n. (The 2n+1 is there to denote that we need odd multiples of pi/2) or alternatively you could write
\[x = 2n \pi \pm \frac{ \pi }{ 2 }\]
Both are acceptable.

Thus the domain of secant is all real x EXCEPT when \[x = (2n+1) \frac{ \pi }{ 2 }\] for some integer n.

The range of secant is defined by the relative minimum and maximum.
As mentioned before, the maximum of cosine become a relative min y =1
and the minimum of cosine becomes a relative maximum at y = -1
The function will increase ahead of y = 1 and decrease below y = -1
Thus the range is (-infinity, -1] and [1, infinity) or
\[y \le -1; y \ge 1\]