Doubt about the power of $x$ of the function to find that $f(x) = 3$ is an even function

Question

Using the definition of even functions, $f(-x) = f(x) = 3$, hence it is an even function.

Also, it is being mentioned that if $f(x)$ is an even power of $x$, then it is an even function of $x$. Same for an odd function.

We know that the equation of a horizontal line in slope-intercept form is $y = mx + b$ where $m$ is the slope and $b$ is the $y$-intercept.

That means, the equation of the horizontal line $y = 3$ can be written as,

$y = 0x^1 + 3$

where $m = 0$ and b = 3.

Therefore,

$y = f(x) = 0x^1 + 3 = 0x^1 + 3x^0$

Now, what is the power of the function above, $0$ or $1$?

I want to apply the even power method to determine whether $f(x) = 3$ is an even function or let any function having multiple terms with $x^n$ to determine whether it is an even function (or an odd function let's say) just by using the power method.

Answer 1

Generally, we say the degree of a polynomial is given by the largest power of $x$ with a nonzero coefficient.

Technical Aside: In relevant contexts, this is often given a few caveats (probably beyond the scope of this thread). For instance:

• The polynomial should be written in the standard form $a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$.
• If you do not like the choice of $a_0 x^0$ as a valid interpretation of the constant coefficient (to ensure that constant functions are of degree $0$), you would have to insist upon it by convention, i.e. define from the outset that the degree of such polynomials is $0$ (or use a clunky definition that ultimately is just an obfuscation of such).
• The degree of the all-zero polynomial is somewhat contentious as an additional special case of the previous bullet; I've seen people give it degrees of $0$ and $-\infty$ in the past; others evidently just do not assign it one whatsoever.

Hence, $x^2$ and $x^2 + 0x^3 + 0x^4 + 0x^{100}$ are both degree-$2$ polynomials.

We interpret the constant-function case as a polynomial of degree $0$ (hence even).

Answer 2

the power of the function 0^1+3^0 is 0- since it's the one with a nonzero coefficient. So when checking for any polynomial eqn- remember to take into account their coefficients- they have to be nonzero