Odd/Even Functions: Check Symmetry over Y Axis First

In summary, it is necessary to first check for symmetry over the y-axis before determining if a function is odd or even. If the function is not symmetric over the y-axis, then it is not necessary to check for odd or even symmetry.
  • #1
Yankel
395
0
Hello

I have a theoretical question. When I check if a function is odd or even, I usually check:

f(x)=f(-x) or f(-x)=-f(x)

someone told me today that before checking it, I first need to check the symmetry over the Y axis, and if the function is not symmetric over Y, there is no point of checking for odd or even.

Can someone explain this to me, and give a simple example of how to check for symmetry ?

thanks !
 
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  • #2
If f(-x) = f(x), then it is symmetric about the y-axis, i.e., it is even. I think your method is best.
 
  • #3
Yankel said:
Hello

I have a theoretical question. When I check if a function is odd or even, I usually check:

f(x)=f(-x) or f(-x)=-f(x)

someone told me today that before checking it, I first need to check the symmetry over the Y axis, and if the function is not symmetric over Y, there is no point of checking for odd or even.

Can someone explain this to me, and give a simple example of how to check for symmetry ?

thanks !
Think about the transformations to f(x) represented by f(-x) and -f(x) ...
 

FAQ: Odd/Even Functions: Check Symmetry over Y Axis First

What are odd and even functions?

Odd and even functions are two types of mathematical functions that exhibit symmetry over the y-axis. An odd function is a function that satisfies the property f(-x) = -f(x), meaning that its output values are symmetric about the origin. On the other hand, an even function is a function that satisfies the property f(-x) = f(x), meaning that its output values are symmetric about the y-axis.

How can I tell if a function is odd or even?

To determine if a function is odd or even, you can use the symmetry test. If the function satisfies the property f(-x) = -f(x), it is an odd function. If the function satisfies the property f(-x) = f(x), it is an even function. Another way to determine the symmetry of a function is by graphing it. An odd function will have a rotational symmetry of 180 degrees about the origin, while an even function will have a reflective symmetry about the y-axis.

Why is it important to check for symmetry over the y-axis first?

Checking for symmetry over the y-axis first is important because it is the most basic and fundamental type of symmetry. If a function is symmetric about the y-axis, it is either an odd or an even function. This step helps us to quickly identify the type of function we are dealing with and simplifies the process of determining its properties.

Can a function be both odd and even?

No, a function cannot be both odd and even. A function can only have one type of symmetry, either odd or even. A function that is both odd and even would have to satisfy the properties f(-x) = -f(x) and f(-x) = f(x) at the same time, which is not possible.

How are odd and even functions used in real life?

Odd and even functions have many applications in the real world, especially in physics and engineering. For example, odd functions are used to describe physical phenomena such as electric fields and magnetic fields. Even functions are used to describe symmetric structures such as bridges and buildings. In mathematics, odd and even functions are used to solve equations and simplify complex functions.

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