Over which of the following domains is f(x) = csc(0.1x + 1.2) defined at all points and invertible?
a. x = [0,10]
b. x = [10,20]
c. x = [20,30]
d. x = [30,40]
I know it's either a or d since 20 is an asymptote, I'm just not sure which one.
its c, i took the practice quiz thing
To determine over which domain the given function, f(x) = csc(0.1x + 1.2), is defined at all points and invertible, we need to consider two conditions:
1. The function must be defined for all x-values within the domain.
2. The function must be one-to-one, meaning it must pass the horizontal line test, for it to be invertible.
The cosecant function, csc(x), is defined for all real numbers except when the sine function, sin(x), is equal to zero. The sine function is zero at x = 0, π, 2π, 3π, and so on.
For the given function, the argument inside the cosecant function is 0.1x + 1.2. To find the values of x for which sin(0.1x + 1.2) = 0, we set 0.1x + 1.2 = 0 and solve for x:
0.1x + 1.2 = 0
0.1x = -1.2
x = -12
Therefore, the function is not defined when x = -12, as the sine function is zero at that point.
Now let's consider the given options:
a. x = [0,10]
In this domain, x does not include -12, so the function is defined at all points. However, since -12 is not in this domain, the function is still invertible.
b. x = [10,20]
In this domain, x does not include -12, so the function is defined at all points. Again, since -12 is not in this domain, the function is invertible.
c. x = [20,30]
In this domain, x includes -12, so the function is not defined for all x-values within this interval. Therefore, it is not invertible.
d. x = [30,40]
Similar to option c, this domain includes -12, so the function is not defined for all x-values within this interval. Hence, it is not invertible.
Based on the analysis, the function f(x) = csc(0.1x + 1.2) is defined at all points and invertible over the domain x = [0,10]. Therefore, the correct answer is option a.
To determine the domain of a function and its invertibility, we need to consider any restrictions on the function. In this case, the given function is f(x) = csc(0.1x + 1.2).
The cosecant function, csc(x), is defined as the reciprocal of the sine function, sin(x). We know that the sine function has vertical asymptotes at x = (kπ + π/2), where k is an integer. In the case of the cosecant function, the vertical asymptotes occur at the zeros of the sine function, so x = kπ.
In the given function f(x) = csc(0.1x + 1.2), the term inside the cosecant function is (0.1x + 1.2). To find the vertical asymptotes, we set this term equal to kπ:
0.1x + 1.2 = kπ
Now, we need to solve for x to determine the values of x where the function is undefined. Rearranging the equation, we have:
0.1x = kπ - 1.2
x = (kπ - 1.2) / 0.1
So, the expression (kπ - 1.2) / 0.1 gives us the values of x where the function is undefined, i.e., the vertical asymptotes.
Now, let's check the options given:
a. x = [0,10]: The range of x in this interval does not include any multiple of π, so there are no vertical asymptotes within this interval.
b. x = [10,20]: In this interval, the value of x = 20 is the vertical asymptote, which corresponds to (kπ - 1.2) / 0.1 for some k.
c. x = [20,30]: In this interval, the value of x = 20 is the vertical asymptote, which corresponds to (kπ - 1.2) / 0.1 for some k.
d. x = [30,40]: The range of x in this interval does not include any multiple of π, so there are no vertical asymptotes within this interval.
Hence, the correct option is either a or d.
It is not (D) because f(x) has a maximum at 35.12
That means that f(x) fails the horizontal line test on the interval (30.24,40), so it is not invertible there.