Even & Odd Functions: What Happens When Applied?

In summary, when applying an even function to another even function, the end result will still be even. The same is true for odd functions. However, if an even function is applied to an odd function, the result cannot be determined without knowing the specific functions involved.
  • #1
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If i had an even fuction and i applied another even function to it, would the end result still be even?

Other questions follow the same pattern, if i had an odd function and applied another odd function to it, would it still be odd?

And suppose i had an even function and applied an odd function to it, what would it be then?

I suppose the first question is true, the next two i had simply no idea, I can't tell what would happen to functions with many terms in if i spplied other functions to it. I'm not after to prove it, because I'm aweful at that :P But something to help me figure out the answer to these questions would be very helpful!

Thanks
 
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  • #2
A function is even if f(-x)=f(x) and odd if f(-x)=-f(x). Write the composition as f(g(x)). Use whatever the rules are for f and g to compare f(g(-x)) to f(g(x)).
 
  • #3


I would like to clarify the concept of even and odd functions before answering your questions. An even function is a mathematical function that satisfies the property f(x) = f(-x). This means that if you plug in the opposite of a number into the function, you will get the same output. Examples of even functions include f(x) = x^2 and f(x) = cos(x). On the other hand, an odd function is a mathematical function that satisfies the property f(x) = -f(-x). This means that if you plug in the opposite of a number into the function, you will get the opposite of the output. Examples of odd functions include f(x) = x^3 and f(x) = sin(x).

Now, to answer your first question, if you have an even function and you apply another even function to it, the end result will still be even. This is because both functions have the same symmetry, so when you apply one to the other, the symmetry is preserved.

Similarly, if you have an odd function and you apply another odd function to it, the end result will still be odd. This is because both functions have the opposite symmetry, so when you apply one to the other, the symmetry is again preserved.

However, if you have an even function and you apply an odd function to it, the end result will not necessarily be even or odd. It will depend on the specific functions that are being applied. For example, if you apply f(x) = x^2 to f(x) = sin(x), the end result will be neither even nor odd. It will be a combination of both even and odd properties.

To better understand what happens when applying functions to each other, you can graph the functions and observe the changes in their symmetry. You can also try plugging in different values for x to see how the output changes.

In conclusion, when applying even and odd functions to each other, the resulting function will have the same symmetry as the original functions. However, when applying an even function to an odd function, the resulting function may not have a specific symmetry. I hope this helps in understanding the concept of even and odd functions and their behavior when applied to each other.
 

1. What is an even function?

An even function is a mathematical function that satisfies the property f(x) = f(-x). This means that the value of the function at a certain input x is equal to the value of the function at the opposite input -x.

2. What is an odd function?

An odd function is a mathematical function that satisfies the property f(x) = -f(-x). This means that the value of the function at a certain input x is equal to the negative value of the function at the opposite input -x.

3. What happens when an even function is applied to an even number?

When an even function is applied to an even number, the output will also be an even number. This is because the input and its opposite will have the same value, satisfying the property f(x) = f(-x).

4. What happens when an odd function is applied to an even number?

When an odd function is applied to an even number, the output will be an odd number. This is because the input and its opposite will have opposite values, satisfying the property f(x) = -f(-x).

5. Can a function be both even and odd?

No, a function cannot be both even and odd. This is because the properties of even and odd functions are mutually exclusive. A function can only satisfy one of the properties, not both.

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