I am struggling solving the following exercise:
Let $a \in \mathbb{R}^n$ and $ b \in \mathbb{R}$ and define $f : \mathbb{R}^n \rightarrow \mathbb{R}$ by
$f (x)=\langle x,a \rangle + b, x\in \mathbb{R}^n$
Show that for every convex set $X$ in $\mathbb{R}^n$, $ f (X) =\{ f(x):x \in X \} $ is a convex set in $\mathbb{R}$.
Thank you!
Well, if $t_1+t_2=1$ and $t_1,t_2\geq 0$ then for arbitrary Elements $f(x_1),f(x_2)\in f(X)$ $$ t_1f(x_1)+t_2f(x_2)=t_1\langle x_1,a \rangle+t_1b+t_2\langle x_2,a \rangle+t_2b = \langle t_1x_1+t_2x_2,a \rangle + (t_1+t_2)b $$ and so this is $f(t_1x_1+t_2x_2)$ and (since $X$ is convex) also an Element of $f(X)$