Given $g: \mathbb{R}^n \to \mathbb{R}$ is convex and $f:\mathbb{R} \to \mathbb{R}$ is convex and increasing. Show that $(f \circ g): \mathbb{R}^n \to \mathbb{R}$ is convex.
I had no problem proving the previous statement but now I need to use it to show that $h(\mathbf x)=\exp(\mathbf x^T \mathbf x)$ is convex and then show was as counterexample that $k(\mathbf x)=\exp(-\mathbf x^T \mathbf x)$ is not.
I don't know what $\exp(\mathbf x^T \mathbf x)$ means... any help would be greatly appreciated!
If $\mathbf{x}=(x_i)_i$ is a (column) vector, then $\mathbf{x}^T\mathbf{x}=\sum_ix_i^2$ gives you the scalar product of $\mathbf{x}$ with itself (and thus its $2$-norm squared).
$\exp$ is simply the exponential function, $\exp(t)=e^t$.