Question

How do you determine if `f(x)=1/x^2` is an even or odd function?

Answer 1

By substituting (-1x) for all `x`'s in the function, then evaluate the function.

Explanation:

To determine if a function is even, odd or neither, plug in (-1`x`) in place of all `x`'s in the function, then determine if any of the signs in the function have changed from positive to negative, negative to positive, or neither.

Even function: NO sign changes
Odd function: ALL signs change
Neither: some signs change, some signs do not change.

Example: `f(x)=1/x^2`
1. Replace 'all' `x` 's with (-1x):
`f(-1x)=1/(-1x)^2`
2. Check to see if any signs have changed in the function.
Since `(-1x)^2=x^2`, the sign of the function has not changed. So this function is an EVEN function.
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Quick Trick for advanced students:
Even functions: Every `x`-term will have an even exponent, and a constant term may or may not exist. (`f(x)=x^4-x^2+2` is an even function.

Odd functions: Every `x`-term will have an odd exponent and a constant term will not exist. `f(x)=x^5+x^3+x` is an odd function.

Neither: A combination of both even and odd exponents will exist or at least one odd exponent will exist as well as a a constant term. `f(x)=x^5-x^4` and `f(x)=x^5-x^3-2` are neither even nor odd.