This question is from Larson's Precalculus book.
Prove that a function of the form $$y = a_{2n}x^{2n} + a_{2n−2}x^{2n−2} + . . . + a_2x^2 + a_0$$ is an even function.
I don't understand how that form of function is an even function?
Suppose that all coefficients are $1$ and $x=1$ and $n=2$. Then
$$y = a_4x^4 + a_2x^2 + a_0.$$
Because all coefficients are $1$ and $x=1$,
$$y = 1 + 1 + 1 = 3,$$ which is odd.
Could you please explain how am I wrong?
A function $f:D \to \mathbb{R}$ is even if $$f(-x)=f(x), \forall x \in D.$$
So, if $k \in \mathbb{N}$ is an even number, the function $x \mapsto x^k$ is even, since $$(-x)^k=(-1)^kx^k=x^k,$$ and so the functions of the form $g(x)=cx^k$, where $c$ is a constant, is even.
Note that every exponent of the terms of the polynomial are even, so each term is even. Finally, since the sum of even functions is even, the results follows.