How I can to determine the domain and the maximum and minimum local of the following function: f(x)=6sec((pi*x)/8). And I need to graph the function.
Since the secant is the reciprocal of the cosine curve, consider the function
y = 6cos[πx/8]
This reaches a max of 6 and a min of -6
the period is 2π/(π/8) = 16
The max of this cosine curve is (0,6) and (16,6)
the min values are (8,6)
the zeros of the cosine curve result in vertical asymptotes for the secant curve.
zeros of our cosine curve are (4,0) and 12,0)
so sketch vertical asymtotes at x = 4 and x = 12, (you might also do x = -4)
Then sketch the typical U shapes of the secant curves.
The max of the cosine becomes the min of the secant, and the min of the cosine becomes the max of the secant
So the domain is all values of x
the range is :
f(x) ≥ 6 OR f(x) ≤ -6
about 1/4 of the way down the page of this website you see a good illustration of what I have done
http://jwilson.coe.uga.edu/emt668/EMAT6680.2000/Umberger/EMAT6690smu/Day6/Day6.html