Hello everyone, I hope you are all having a fantastic day.
I had a question regarding even and odd functions. I understand that even functions must have each exponent as even, and odd functions must have each exponent as odd, however, what do you do in the case of having a mixed function, for instance: (x^2 + 1)/(x+1), or any other function that you cannot factorize. Is the only solution to plug in values, or is there a trick I am missing.
Have a great day!
You can use the definition of an even/ odd function:
If f is an even function then:
f(-x) = f(x)
Similarly, if f is an odd function then:
f(-x) = -f(x)
As for the function, you posted:
f(x) = \frac{x^2 + 1}{x + 1}
It’s neither even or odd because f(-x) is neither f(x) or -f(x).
Sure, and I understand the definition. I supppose what I was getting at was for the more complex versions of f(x), is the only way to ascertain if it is even or odd (without it blatantly only having only odd or even exponents), to plug in numbers and see if f(x) = f(x) or -f(x)?
They’re not specific numbers, are they? It’s more like you’re showing that the symmetry holds for all real values for x.
Anyway yes, you’d check with the definition and if it works then it works. It’s the most rigorous check to do and isn’t that time confusing either.
Understood, thank you!