Question

If f(2x) + 3f(1/2x) = x+1 and x≠0 , then the value of 4f(2) equals

Answer 1

$f(2x)+f(\frac{1}{2x})=x+1$Let $u=2x\\\Rightarrow x=\frac{u}{2}\\\Rightarrow f(u)+3f(\frac{1}{u})=\frac{u}{2}+1$Since, u is a dummy variable, we can replace u by x.$\\f(x)+3f(\frac{1}{x})=\frac{x}{2}+1$Replacing x by $\frac{1}{u}$ and then replacing u by x we get,$\\f(\frac{1}{x})+3f(x)=\frac{1}{2x}+1$Consider the above two equations.$\\\begin{cases}f(x)+3f(\frac{1}{x})=\frac{x}{2}+1\\f(\frac{1}{x})+3f(x)=\frac{1}{2x}+1\end{cases} \\eq 1 - 3 \times eq 2\\-8f(x)=\frac{x}{2}-\frac{3}{2x}-2\\\Rightarrow f(x)=\frac{3}{16x}+\frac{1}{4}-\frac{x}{16}$Put x=2$f(2)=\frac{3}{32}+\frac{1}{4}-\frac{1}{8}\\\Rightarrow 4f(2)=\frac{3}{8}+1-\frac{1}{2}\\=0.875$