Domain and Range of Secant. Help Please?

Question

Find the Domain and Range of

f(t) = sec(pi*t/4)

Please explain, i'm trying to grasp it.
I know I can find the range and domain by graphing it with my Ti - 89, but how do i figure it out algebralicly.

Thanks


Heres what the solution manual shows as the answer, could you explain what he is doing there? Im lost.
Thanks

http://img207.imageshack.us/my.php?image=mathprob2uf9.jpg

Answer 1

Find the domain and range of:

f(t) = sec(πt/4)

Secant is the reciprocal of cosine. Secant will have a vertical asymptote everywhere that cosine equals zero. Those will be the only points that are NOT in the domain.

sec(πt/4) = 1/cos(πt/4)

Period = 2π/(π/4) = 8
cos(πt/4) = 0 at t = 2, 6, etc.
t = 2 + 4k, where k is an integer

So the domain of sec(πt/4) is:

All the real numbers except t = 2 + 4k, where k is an integer.
_____________

The range of cosine is:

-1 ≤ cosine ≤ 1

Since secant = 1/cosine its range is:

| secant | ≥ 1

For the specific function f(t) = sec(πt/4) the range is:

(-∞, -1] U [1, ∞)

Answer 2

Start by thinking about the definition of Sec, Sec=1/cos. So f(t)=1/cos(pi*t/4). Now think about the cos function. The domain of cos is all t and the range of cos is [-1,1]. Now think what happens when you divide 1 by numbers in the range of cos.

You will get values close to +/- infinity as cos gets close to 0 from either side. You will get values of +/-1 when cos is +/- 1. You will never get values in (-1,1) bedcause this would mean that cos would have to be greater than one. Combining these facts you get the range to be ( -inf,1] U [1,inf).

For the domain, think where is 1/cos undefined. It is undefined when you divide by 0. So solve for cos(pi*t/4)=0 to get the excluded points on the domain. cos(x)=0 when x=(2*k+1)*pi/2. So you have to solve pi*t/4=(2*k+1)*pi/2 which is the first bit of working in the link. The domain is all t except these points, which we usually express using the not equal to sign.

Graphing is not a bad way to start with figuring our domain and range, this one you could (hopfully) sketch by hand by sketching cos first then building up to Sec.

Answer 3

well as secant is the reciprocal of cosine... the domain is all the real values except those for which cosine (pi*t/4)=0;
now cos (pi*t/4)=0=cos (pi/2) {and all odd multiples of (pi/2)}

please check with your Ti that all odd multiples of pi/2 has cosine = 0;

now odd numbers are denoted by (2k+1) and by putting integral values for k you will get all odd numbers........

therefore;
0=cos((2k+1)*pi/2);
0=cos(pi*t/4)
equate and remove cosine.....
pi*t/4 = (2k+1)*pi/2;
t = 2(2k+1);

this means that all values are acceptable for t in order that sec is defined except for t= 2(2k+1) where k=1,2,3.......

im sorry but i cant help you with the range part......