What domain makes the function y=csc(x) invertible?
[0, pi]
[-pi/2,0) u (0,pi/2]
[-pi/2,pi/2]
[0,pi/2) u (pi/2, pi]
How do you show your work on this?
you have to avoid the asymptotes, and make sure cscx is always increasing or decreasing.
So, B looks like your best bet.
Think of the graph of sin(x)
To determine the domain that makes the function y = csc(x) invertible, we need to find the interval where the function is one-to-one, meaning each input value has a unique output value.
The reciprocal of the sine function, csc(x), is defined as 1/sin(x). Since the sine function has a range of (-1, 1), the reciprocal, 1/sin(x), will have a range of (-∞, -1] ∪ [1, ∞).
To find the domain that makes the function y = csc(x) invertible, we need to identify the interval for x such that the range of y falls within (-∞, -1] ∪ [1, ∞).
Let's analyze the given options:
[0, pi]: This interval includes all values from 0 to pi. However, csc(x) is not defined for x = 0 since it results in division by zero. Therefore, this interval is not valid.
[-pi/2,0) U (0,pi/2]: This interval includes all values from -pi/2 to 0, excluding 0, and all values from 0 to pi/2. Thus, it covers the entire range of csc(x) and does not include any values where csc(x) is undefined. This makes the function invertible. Therefore, this interval is valid.
[-pi/2,pi/2]: This interval includes all values from -pi/2 to pi/2 but does not exclude 0. Since csc(x) is undefined for x = 0, this interval is not valid.
[0,pi/2) U (pi/2, pi]: This interval includes all values from 0 to pi/2 (excluding pi/2) and all values from pi/2 to pi. However, it does not include the negative values of the range of csc(x), which are required to make the function invertible. Therefore, this interval is not valid.
Thus, the correct answer is [-pi/2,0) U (0,pi/2], which ensures that x is within the appropriate range for csc(x) to be invertible.
To show your work for this question, you can simply explain the reasoning mentioned above, stating why each option is valid or not valid based on the properties of the csc(x) function and its range.