Finding a value of a function continuous at an other point

Question

Let $f : \mathbb{R}\to\mathbb{R}$ be a function which is continuous at $0$ and $f(0) = 1$. Also assume that $f$ satisfies the following relation for all $x$: $$f(x) − f\left(\frac x2\right) = \frac{3x^2}4+x$$ Find $f(3)$.

(I'm unable to even start.. any hint is appreciated)

Answer 1

$f(x)-f(x/2)=3x^2/4 +x$

$f(x/2)-f(x/4)=3x^2/16 +x/2$

$f(x/4)-f(x/8)=3x^2/64 +x/4$

$\cdot$

$\cdot$

$\cdot$

$f(x/2^n)-f(x/2^{(n+1)})=3x^2/2^{(2n+2)} +x/2^n$

Now as $n \rightarrow \infty$ , we get,

$f(x) - f(0) = \frac{3x^{2}(1+1/4+...)}{4} + x(1+1/2+1/4+....)$

$f(x)-1=x^2 +2x$

$f(x)= (x+1)^2$

$f(3)=16$