Question

True or False: For a trigonometric function, y = f(x), then x = F-1(y). Explain your answer.

Answer 1

If I'm interpreting this correctly, the statement is saying that in a trigonometric function if y is a certain value for a certain value of x, then x will always be a certain value for a certain value of y.

Answer 2

Now, thinking back to trig functions you've seen before, can you find any examples to support or disprove this statement?

Answer 3

I guess, I am not really understanding this

Answer 4

Ok, so there's a test you can do to see if this statement is true or not. If you draw a horizontal line through the function, if the statement is true, the horizontal line should only pass through the curve of the function once. If the horizontal line passes through the function more than once, you know that any particular y value can match with many different x values, which disproves the statement.

Answer 5

It is passed once , it think :/

Answer 6

SO true because it passed the horizontal line test and pass through the curve of the function once

Answer 7

Ok, just checking, what function did you use the horizontal line test on?

Answer 8

first one

Answer 9

y = f(x),

Answer 10

Ah, gotcha. So, y=f(x) isn't really a specific function. It's just defining what a function needs to be: for every value of x in the domain there is 1 and only 1 value of y.
You're going to want to test a specific function, like, particularly a trigonometric equation. These are equations that use trigonometric terms, like sine, cosine and tangent. You could use the cosine function from the question we did earlier, for instance.

Answer 11

Okay so y = cosx

Answer 12

Sure. How does the horizontal line test work for y = cos(x)?

Answer 13

nope it fails it

Answer 14

Great! So that means we've disproved the statement.

Answer 15

So false, because it does not pass the horizontal line test. It passes through the curve of the function more than once

Answer 16

Yup. Sounds pretty good

Answer 17

are you going to bed now?

Answer 18

Nah, I'll be up for a while.

Answer 19

True or False: For a one-to-one function, y = f(x), then x = f-1(y). Explain your answer.

Answer 20

Ok, this one is pretty easy to explain. A one-to-one function, by definition, has one y value for every y value and one x value for every y value. No more, no less.

Answer 21

So true

Answer 22

and the explanation part is the defnition of what a one to one function is

Answer 23

Yup, you got it

Answer 24

I think that'd work, yes

Answer 25

True or False: For any function, x = f-1(y), then y = f(x). Explain your answer.

Answer 26

For this one, you can think of it as similar to the first question in this series, only instead of "every y-value only has one x value" it's "every x value has only one y-value:"

Answer 27

So it owld be the same answer right

Answer 28

By the definition of a function this has to be true. You can do the vertical line test to demonstrate this, but you know enough about functions to accept it at face value, I think.

Answer 29

Except we would use the vertical line test

Answer 30

Yup

Answer 31

Oh, and you can use any function, not just trig functions

Answer 32

True, because it passes the vertical line test. It does not pass through the curve of the function more than once.

Answer 33

I completely agree.

Answer 34

We can actually take a look at a the sine function to see whether this one is true or not

Answer 35

no it is not

Answer 36

because it fails the horizontal line test

Answer 37

or wait its the vertical line test right

Answer 38

So, this one is a little different from the others. It's asking about the direction of the line over a certain interval. So, between the two intervals mentioned, pi/2 and 3pi/2, does the function increase or decrease?

Answer 39

increase

Answer 40

Yes, you got it!

Answer 41

So true, because the direction of the lines over the two certain intervals are increasing within each other

Answer 42

Exactly

Answer 43

Going to bed now??

Answer 44

Heheh, no, I don't have classes until late tomorrow. College is weird like that.

Answer 45

Lol i bet

Answer 46

Explain the meaning of y = cos^-1x.

Answer 47

So, whenever anything is taken to the power of negative 1, it means you are getting the reciprocal of that value. For instance, 2^(-1) is the same as 1/2. 5^(-1) is the same as 1/5. in the same way, Cos(x)^-1 is the same as 1/cos(x)

Answer 48

It's difficult to tell with the notation here, but I think that is what you meant?

Answer 49

I screenshot it so you can see

Answer 50

Ok, so I did understand correctly. That's good. I stand by what I said.

Answer 51

Okay so the meaning of y = cos^-1x is whenever anything is taken to the power of negative 1, it means you are getting the reciprocal of that value. From this function given, we can see that Cos(x)^-1 is the same as 1/cos(x)

Answer 52

Yup, I think you understand.

Answer 53

Thsi would be true

Answer 54

If i remember correctly, when a function is undefined, it is undefined on an interval or at certain points. whether and/or where a function is undefined depends on the given domain of the function.

Answer 55

That's interesting. Keep in mind that sec(x) is equal to 1/cos(x). So, Sec(x)^-1, would be 1/sec(x), which is actually the same as cos(x). I would say that cos(0.5) is defined, so I would say the statement is false.

Answer 56

could you check this over

Answer 57

I'd say your math checks out.

And I also want to make a correction to what I said earlier.
With plain numbers, what I said about something to the power of (-1) is true. In this case, it means you're taking the inverse of the function. You can think of this as moving backwards; usually, you'd convert x, the angle or arc length, to y, the value of, say, cosine for that particular value. With an inverse function, you're given the value of cosine and use the function to find the value of the angle or, in this case the arc length.
Even with my wrong explanation, I'm impressed you got these right!

Answer 58

I think now is when I'm going to start going to sleep. Anything else before I head out?

Answer 59

Nope, thats all Just to make sure even with your wrong explanation is there any specific question I need to correct, or are my answers still correct

Answer 60

Nah, you got it right

Answer 61

Nvm, you can go to bed we can continue tomorrow or something. I don't want to hold you up

Answer 62

Goodnight, and thank you for all your help.. I really do appreciate it

Answer 63

No worries. See you around!