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Advanced Math/Period, Domain, Range of Function
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Question
1. y = 3csc(3x + π) - 2
2. y = 2 sin(4x + π) + 3
Determine
a. the period
b. the domain
c. the range
period (how often it repeats)
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The period for csc(x) and sin(x) are the same, since csc(x)=1/sin(x).
The period for sin(z) is 2π. For 1, the period need to be divided by 3 since that is the multiple of x, so its 2π/3. For the second one, we have a 4 instead of a 3, so its 2π/4 = π/2.
domain=plug in
range=get out
domain (what you can put in)
----------------------------
The domain for csc(z) is the same as the domain for 1/sin(z) with the exception of points where sin(z) is 0. sin(nπ) is zero for all even n, so csc(nπ) is undefined for all even n. The domain for csc(z) would then be all z from (-∞,∞) except for nπ where n is an odd integer (positive or negative). For the domain of this problem, we need 3x+π to the same as nπ where n is odd, so x needs to be nπ where n is even. So the domain would be (-∞, ∞) without all of the points nπ where n is any even integer (positive or negative).
The domain of sin(z) is (-∞, ∞). For any multiple of z, the domain is still the same. For any linear function, it is still the same. This is true whether the multiples are on the inside or outside. For this problem, then, the domain is (-∞, ∞).
range (what comes out of the function)
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The range for csc(z) 1/range for sin(z). That would make the range for csc(z) to (-∞,-1] amd [1,∞). Since the csc in the problem is multiplied by 3, the new range is (-∞,-3] and [3,∞).
The range for sin(z) is between sin(-π), which is -1, to sin(π), which is 1, so that it's [-1,1]. The range for sin(ax+b) (where z=ax+b) is the same, since a and b are on the inside. In this problem, note that its c*sin(z)+d, so the range becomes [d-c, d+c].
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Scott A Wilson
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