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A function is even if `f(-x)=f(x)` :
(a) `f(x)=3x^4+4` `f(-x)=3(-x)^4+4=3x^4+4` even
(b) `f(x)=x^3+4` `f(-x)=(-x)^3+4=-x^3+4`
(c) `f(x)=x^2+x-3` `f(-x)=(-x)^2+(-x)-3=x^2-x-3` not even
(d) `f(x)=2x^4+x^2+1` `f(-x)=2(-x)^4+(-x)^2+1=2x^4+x^2+1` even
(e) `f(x)=4` `f(-x)=4` even
(f) `f(x)=x^4+x^3+x^2+x+1` `f(-x)=(-x)^4+(-x)^3+(-x)^2+(-x)+1=x^4-x^3+x^2-x+1` not even
Note that for polynomials, it suffices that all exponents be even. Also note that not even does not mean odd.
For every function marked as even, the function is symmetric about the y-axis. Here are their graphs:
If we have a function where f(x) = f(-x), the function is said to be even.
In mathematical terms, even functions are said to be reflexive (or symmetric) about the y-axis.
Therefore, in order to test whether a function is even or not, put –x for x in f(x)and test whether f(x) is returned or not.
When all the exponents of x in f(x) are even (including 0), the function is even. If any of the exponents is odd, the function is not even.
Thus in example a),
f(x) = 3x^4 + 4
f(-x) = 3(-x)^4 + 4 = 3x^4 + 4 = f(x);
Hence it is even.
Testing in similar arguments it is clear that functions a), d) and e) are even.
In your answer you say that if a polynomial has a term of odd degree then the polynomial is odd. This is incorrect. Not even does not imply odd.
An odd function is a function where f(-x)=-f(x). The graph has 180 degree rotational symmetry about the origin. A polynomial that has every term of odd degree (specifically no constant term) is odd, but a function like `x^3+x^2` is neither even nor odd.
Thanks for the correction. Here is the corrigendum: "When all the exponents of x in f(x) are even (including 0), the function is even. If any of the exponents is odd, the function is not even."
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