Finding parity of unknown function

Question

I'm working through the problem set for MIT 18.01 on Single Variable Calculus, but I'm stuck on one of the first problems. It reads:

Identify each of the following as even, odd , or neither.

and the problem in question is

e) $J_0(x^2)$, where $J_0(x)$ is a function you never heard of.

I solved the previous problems by inputting values of x into the equation and finding the parity based off of the premises that:

• A function $f$ is even if and only if $f(x)=f(-x)$
• A function $f$ is odd if and only if $f(x)=-f(-x)$

I don't understand how I can apply these to a function without knowing how to evaluate the function. I'm obviously missing something, so please enlighten me. Thanks,

Answer 1

Just plug in $x$ and $-x$ into the function

$$h(x) = J_0(x^2)$$ $$h(-x) = J_0((-x)^2) = J_0(x^2)$$

Therefore, the function is even

Answer 2

HINT

Notice that if $f(x) = J_0(x^2)$ then $f$ is a composition of some function $J_0$ with a very special function $x^2$. Is $x^2$ odd or even?

Can you see what happens if you plug in and simplify $f(-x)$?