Solution at $(z+1)f(x) - zf(-x) = x^4$

Question

I need your help on $(z+1)f(x) - zf(-x) = x^4$

a) Prove f is even.

b) Find $f(x).$

Can you help me?

Answer 1

a) Plug in $-x$ for $x$: $$ (z+1)f(-x)-zf(x)=x^4$$ and subtract from the original equation $$ f(x)-f(-x)=0$$

b) Substitute $f(-x)$ for $f(x)$: $$ f(x)=x^4$$

Answer 2

$(z+1) f(x) - zf(-x) = x^4\\ z(f(x) - f(-x)) + f(x) = x^4$

The right hand side is even. Which means that the left hand side is also even.

$z(f(x) - f(-x)) + f(x) = z(f(-x) - f(x)) + f(-x)$

Collect the $f(x)$ terms on one side and the $f(-x)$ terms on the other

$(2z+1)f(x) = (2z+1)f(-x)\\ f(x) = f(-x)$

$f(x)$ is even

$z(f(x) - f(-x)) + f(x) = x^4\\ f(x) = x^4$