Answer
(a) The function is even if for every $x$ from its domain $f(-x)=f(x)$. If the graph of the function is symmetric with respect to $y$ axis then the function is even. The examples are $f(x)=x^2$, $f(x)=\cos x$, $f(x)=2x^4+1.$
(b) (a) The function is odd if for every $x$ from its domain $f(-x)=-f(x)$. If the graph of the function is symmetric with respect to the origin then the function is even. The examples are $f(x)=x$, $f(x)=\sin x$, $f(x)=x^3.$
Work Step by Step
(a) The function is even if for every $x$ from its domain $f(-x)=f(x)$.
If the graph of the function is symmetric with respect to $y$ axis then the function is even. This is because whatever "happens" to the function at $x$, the same will happen at $-x$.
The examples are $f(x)=x^2$, $f(x)=\cos x$, $f(x)=2x^4+1.$
This is because
$$(-x)^2=x^2,\quad \cos{-x}=\cos x,\quad 2(-x)^4+1=2x^4+1$$
(b) (a) The function is odd if for every $x$ from its domain $f(-x)=-f(x)$.
If the graph of the function is symmetric with respect to the origin then the function is even. This is because if the function passes through the point $(x,y)$, it will pass through the point $(-x,-y)$.
The examples are $f(x)=x$, $f(x)=\sin x$, $f(x)=x^3.$ This is because
$$(-x)=-x,\quad (-x)^3=-x^3,\quad \sin(-x)=-\sin x.$$
