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1. consider the function f(x)=cot x . which of the following are true? select all that apply
a. it is an even function
b. has an asymptote at x=0
c. has a zero at x=0
d. has a period of pi.
3.look at the graph of f(x)=sin 1/x. is the graph periodic? why or why not?
a. yes it is periodic. the value of f(x) varies between -1 and 1 repeatedly
b. yes it is periodic. its a transformation of sin (x)
c. no. it does not repeat on regular intervals
d. no. it is a transformation of 1/x
what set describes the zeroes of the function f(x)=6pi sin (4x) shown in the graph
a. x|x=k pi for every integer k
b. x|x=4kpi for every integer k
c. x|x=k/4 pi for every integer k
d. x|x=6k for every integer k

!

1. B & D are TRUE
3. C. No. It does not repeat on regular intervals 
5. C.x|x=k/4 pi for every integer k

1. f(x)=cot x
First, plot a graph of cot(x) using whatever tools you desire. If you do that, you'll see several properties of the graph that make most of the answers rather obvious. Let's see what options are true.
A. It is an even function (FALSE)
A function is even iff f(-f) = f(x) for all x. So let's test that.
cot(pi/4) = 1
cot(-pi/4) = -1
1 is not equal to -1, so the function is NOT even.

B. Has an asymptote at x=0 (TRUE)
cot(x) is defined as cos(x)/sin(x). At x=0, cos(x) will be 1. But sin(x) will be
approaching 0 as x approaches 0. But if x is approaching 0 from negative values, sin(x) will be negative. And is x is approaching 0 from positive values, sin(x) will be positive. So cos(x)/sin(x) approaches either positive, or negative infinity depending upon the direction with which x is approaching 0. And that's pretty much the definition of an asymptote. 

C.Has a zero at x=0 (FALSE)
Since option B shows that we have an asymptote at x=0, we can't have a zero at x=0 as well. So this is false. 

d.has a period of pi. (TRUE)
This is obvious from inspection of the graph (you did plot the graph didn't
you?).

3.Look at the graph of f(x)=sin 1/x. is the graph periodic? why or why not? 
Once again, plot the graph and look for yourself. If you actually plot the graph, the answer becomes obvious.

a. yes it is periodic. the value of f(x) varies between -1 and 1 repeatedly 
* Just because it keeps the same range doesn't make it periodic. It also has to have a regular period and this function does not. So this is a bad choice.

b. yes it is periodic. its a transformation of sin (x)
* This function doesn't transform sin(X).And just because you have a transformation of sin(x) doesn't mean the function is periodic. So this is a bad choice. 
c. no. it does not repeat on regular intervals 
* EXACTLY. The function repeats, but doesn't repeat at a regular interval. So this is the correct choice.

d. no. it is a transformation of 1/x
* Just because a function is a transformation of another function doesn't make the transformed function periodic, or non-periodic. So this is a bad choice.

5. what set describes the zeroes of the function f(x)=6pi sin (4x) shown in the graph.
The zeros for the function will occur at each point the sin function returns 0 which is every integer multiple of pi. Now let's look at the available options and see which ones give every integer multiple of pi.

a.x|x=k pi for every integer k
* Yes, this will give only zeros. But it won't give every zero. For instance, the value pi can not be achieved. So this is a bad choice.

b. x|x=4kpi for every integer k
* Same problem applies. Will give zeros, but not every zero. So it's a bad
choice.

c.x|x=k/4 pi for every integer k
* Perfect!, k=0 gives x=0. k=1 gives x=pi, k=2 gives x=2pi, etc. So this is the correct choice in that it gives every possible 0 for each value of k. 

d. x|x=6k for every integer k thanks !
* Same problem. This gives zeros, but not every zero.

For question 1, the correct options are B and D. For question 3, the correct option is C. And for question 5, the correct option is C.

(1). The given function is f(x)=cot x.

It is a trigonometric function which is equal to the ratio of cos x and sin x.

The function can be written as,

.

Now, as x approaches 0, sin x also approaches 0 and cos x approaches 1. So, the function f(x) will approach infinity and it will form an asymptote at x=0.

The function is an odd function because, .

Now, the function repeats itself after an interval of . So, it is a periodic function with a period of .

Thus, options B and D are correct.

(3). The given function is .

The function does repeat its value but the repetition is not by a fixed interval. See the attached image for more details. The function is sine function. So, its value will vary from -1 to +1.

Thus, option C is correct.

(5). The function is .

The graph of the function is attached to the image.

The function is a periodic function and it is repeating its values after every .

So, the period of the function will be,

where k is an integer.

Thus, option C is correct.

For more details, refer to the link:

link