Understanding Even Functions and the Role of Exponential Functions

In summary, the function exp(−x^2 ) is an even function because replacing x with -x still results in the same function, whether x is positive or negative.
  • #1
ZedCar
354
1
Am I correct in thinking exp(−x^2 ) is an even function?

Thanks
 
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  • #2
Yes! Do you see why?
 
  • #3
micromass said:
Yes! Do you see why?

I think it's because if the x was replaced by (-x) you would have

exp(−(-x)^2 ) = exp(-x^2) which is the same as the original function

It's the same either with a +x or -x making it an even function.
 
  • #4
ZedCar said:
I think it's because if the x was replaced by (-x) you would have

exp(−(-x)^2 ) = exp(-x^2) which is the same as the original function

It's the same either with a +x or -x making it an even function.

Indeed! :smile:
 

1. What is an even function?

An even function is a mathematical function where the output remains the same when the input is replaced by its opposite value. In other words, if f(x) is an even function, then f(-x) = f(x) for all values of x.

2. How can I tell if a function is even?

A function can be identified as even if it follows the rule: f(x) = f(-x) for all values of x. This means that when you substitute a number for x, and then substitute the opposite of that number for x, the output remains the same.

3. Are all polynomials even functions?

No, not all polynomials are even functions. A polynomial is only considered an even function if all of the powers of the variables are even. For example, f(x) = x^2 + 4x + 3 is an even function, but f(x) = x^3 + x + 1 is not.

4. Can an even function have an odd degree?

No, an even function must have an even degree. This is because if the degree is odd, then there will be at least one term with an odd power, which will result in an output that is not the same when the input is replaced by its opposite value.

5. What are some real-world applications of even functions?

Even functions are commonly seen in symmetric shapes, such as circles, ellipses, and parabolas. They are also used in physics to describe symmetrical systems, such as the motion of a pendulum. In engineering, even functions are used in signal processing to filter out noise from a signal.

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