Bergeron230
10

# I need some help with this one :( Function f satisfies 2f(x)−f(2−x) = x+ 2 for all x in its domain R. Find analytic form of f(x).

$f(x)=mx+b\\\\2f(x)=2(mx+b)=2mx+2b\\\\f(2-x)=m(2-x)+b=2m-mx+b\\\\2f(x)-f(2-x)=(2mx+2b)-(2m-mx+b)\\\\=2mx+2b-2m+mx-b=3mx+b-2m\\\\3mx+b-2m=x+2\Rightarrow3m=1\ \wedge\ b-2m=2$ $\left\{\begin{array}{ccc}3m=1&/:3\\b-2m=2\end{array}\right\\\left\{\begin{array}{ccc}m=\frac{1}{3}\\b-2m=2\end{array}\right\\\left\{\begin{array}{ccc}m=\frac{1}{3}\\b-2\cdot\frac{1}{3}=2\end{array}\right\\\left\{\begin{array}{ccc}m=\frac{1}{3}\\b-\frac{2}{3}=2\end{array}\right\\\left\{\begin{array}{ccc}m=\frac{1}{3}\\b=2+\frac{2}{3}\end{array}\right\\\left\{\begin{array}{ccc}m=\frac{1}{3}\\b=2\frac{2}{3}\end{array}\right$ $Answer:f(x)=\frac{1}{3}x+2\frac{2}{3}$