Why is this trigonometry function neither odd nor even?

In summary, the conversation discusses the conditions of an odd or even function and how adding a constant can change the parity of a function. It is mentioned that every function can be separated into an even part and an odd part. It is also explained that the sum of an odd and even function is never odd or even, unless one of the functions is the zero function. The conversation concludes with a request for elaboration on this topic.
  • #1
32
0
f(x)=1+sinx
what am I doing wrong here?
1+sin(-x)= 1-sin(x)
 
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  • #2
Nothing wrong--just remember the definition of an odd or even function that you just used on the sin(x) to calculate your parity transform, and think about whether the function f(x) falls into either of those catagories.
 
  • #3
What are the conditions of an odd or even function
if it were even then f(x) = f(-x) or f(x) - f(-x) = 0
1+sinx =/= 1 + sin(-x)
if it were odd, if you rotate it 180 deg about the origin (the 0,0 point) then the graph remains the same so -f(x) = f(-x) or my personal preference f(x) + f(-x) = 0 since it's easier to remember.
1+sinx + 1 + sin(-x) = 2 =/= 0

I don't know any mathematical explanation that explains why a function is neither odd nor even , but I have noticed that when you add a constant to any given function that does not have one (which usually makes them odd or even), it seems to do the trick.
You must add a constant such that it moves the graph along the ordinate or y-axes in this case, which will make symmetry impossible, but..

..then again if we look at f(x) = 1 + cosx we would see it is even, for
1 + cosx = 1 + cos(-x)

From this I would assume adding a constant to an even function yields no change, but adding a constant to an odd function makes it neither odd nor even.

I really don't know, perhaps someone can elaborate.
 
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  • #5
lendav_rott said:
..then again if we look at f(x) = 1 + cosx we would see it is even, for
1 + cosx = 1 + cos(-x)

From this I would assume adding a constant to an even function yields no change, but adding a constant to an odd function makes it neither odd nor even.

I really don't know, perhaps someone can elaborate.


The sum of an odd function and an even function is never odd or even, unless one of the funtions is the zero funtion.

let e(x) be an even function and o(x) be an odd function.

if f(x) = e(x) + o(x) is an even function, than we have

f(-x) = e(-x) + o(-x) = e(x) - o(x) = f(x) = e(x) + o(x). the e(x) terms cancel and you get 2o(x) = 0.

if it is an odd function than

f(-x) = e(-x) + o(-x) = e(x) - o(x) = -f(x) = -e(x) - o(x). the o(x) terms cancel and you get 2e(x) = 0
 

1. Why is this trigonometry function neither odd nor even?

The reason for this is because the definition of an odd function is that f(-x) = -f(x) and the definition of an even function is that f(-x) = f(x). However, trigonometry functions do not follow these definitions, which is why they are neither odd nor even.

2. Can you provide an example of a trigonometry function that is neither odd nor even?

Yes, the sine function (sin(x)) is an example of a trigonometry function that is neither odd nor even. This is because sin(-x) = -sin(x), which is not the same as sin(x). Therefore, the sine function is neither odd nor even.

3. How does the graph of a trigonometry function that is neither odd nor even look like?

The graph of a trigonometry function that is neither odd nor even will not have any symmetry. It will not be symmetric about the origin (like an odd function) or symmetric about the y-axis (like an even function).

4. Are there any real-world applications of trigonometry functions that are neither odd nor even?

Yes, there are many real-world applications of trigonometry functions that are neither odd nor even. For example, the sine function is used in many applications such as sound and light waves, while the cosine function is used in applications such as architecture and engineering.

5. Can a trigonometry function be both odd and even?

No, a trigonometry function cannot be both odd and even. By definition, a function can only be either odd or even, and not both. However, some trigonometry functions may have certain properties that make them behave like odd or even functions in certain cases.

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