Answer
False
Work Step by Step
Consider $T_1:R \rightarrow R,T_1(x)=x \\
T_2: R \rightarrow R, T_2(x)=-x$
Since $T_1$ and $T_2$ are linear transformations and onto, then $(T_1+T_2):R \rightarrow R$.
We obtain: $(T_1+T_2)(x)=T_1(x)+T_2(x)=0$
Thus, $Ker(T_1+T_2)=R \ne \{0\}$, $T_1 + T_2$ is not one-to-one.
