Finding Even/Odd Function in Intervals: the Case of F(x)

In summary, the conversation discusses how to determine whether a function is even or odd, given that it is defined in three or more intervals. The example of a function defined in the interval (a,d) is used to illustrate this concept, with particular focus on the intervals (-8,0), (0,4), and (4,8). The key to determining whether the function is even or odd is to evaluate the function at different points, such as F(-5) and F(5), and observe the values.
  • #1
ajayguhan
153
1
I know what it means by by even, or odd, function. i also know what it means graphically.

My question is how to find a whether a function is even, or odd? if the function is defined in three intervals or more than three intervals.

Consider a function F(x) defined in the interval (a,d)

F(x)= [ p(x) where x belong to the interval (a, b);
q(x) where x belong to the interval (b, c);
r(x) where x belong to the interval (c, d); ]

The particular problem I'm faced with is:

F(x)= [ p(x)=0 where x belong to the interval (-8,0);
q(x)=4 where x belong to the interval (0,4);
r(x)=8 where x belong to the interval (4,8); ]
 
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  • #2
You say you "know what it means by even, or odd, function" so just apply that. In particular F(-5)= 0. F(5)= 8.

That tells you all you need to know.
 

What is an "even function"?

An even function is a type of mathematical function where the input variable (x) and the output variable (f(x)) have the property that f(-x) = f(x) for all values of x. This means that when the input value is negative, the output value is the same as when the input value is positive. In other words, the function is symmetric about the y-axis.

What is an "odd function"?

An odd function is a type of mathematical function where the input variable (x) and the output variable (f(x)) have the property that f(-x) = -f(x) for all values of x. This means that when the input value is negative, the output value is the negative of when the input value is positive. In other words, the function is symmetric about the origin (0,0).

How do I determine if a function is even or odd?

To determine if a function is even or odd, you can use the properties mentioned above. If the function satisfies f(-x) = f(x), then it is an even function. If the function satisfies f(-x) = -f(x), then it is an odd function. You can also use graphical methods, such as graphing the function and checking for symmetry about the y-axis or origin.

Why is it important to know if a function is even or odd?

Knowing if a function is even or odd can provide useful information about the function's behavior. For example, even functions have a line of symmetry about the y-axis, which means that the function's maximum and minimum values occur at the same x-coordinate. Odd functions, on the other hand, have a line of symmetry about the origin, which means that the function's maximum and minimum values occur at x=0. This information can be helpful in solving problems and graphing functions.

Can a function be both even and odd?

No, a function cannot be both even and odd. In order to be an even function, f(-x) must be equal to f(x) for all values of x, which means that f(-x) cannot be equal to -f(x). Similarly, in order to be an odd function, f(-x) must be equal to -f(x) for all values of x, which means that f(-x) cannot be equal to f(x). Therefore, a function can only be one of the two, not both.

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