\displaystyle f,g\in\mathbb{R}^\mathbb{R},\ \alpha,\beta\in\mathbb{R}\\
\begin{array}{lcl}
C\left( \alpha f+\beta g\right)\left( t\right)&=&\left( \alpha f+\beta g\right)\left( t^2+1\right)\\
&=&\left( \alpha f\right) \left( t^2+1\right) +\left( \beta g\right) \left( t^2+1\right)\\
&=&\alpha f\left( t^2+1\right) +\beta g\left( t^2+1\right)\\
&=&\alpha C \left( f\right)\left( t\right) +\beta C\left( g\right) \left( t\right)
\end{array}