Convex functions

Are the following functions convex ?

1. \(f(x)=αg(x)+β\), where \(g:\mathbb{R}^{n}\rightarrow \mathbb{R}\) is a convex function, and \(α,β\) are scalars, with \(α≥0\).

2. \(h(x)=e^{βx^TAx}\), where \(A\) is a positive semidefinite symmetric \(n×n\) matrix and \(β\) is positive scalar.

\(1.\) The function \(f(x)=αg(x)+β\) can be viewed as a composition \(h(g(x))\) of the function \(h(t)=αt+β\), where \(t∈\mathbb{R}\) , and the function \(g(x)\) for \(x∈\mathbb{R}^{n}\). In this case \(h\) is convex and monotonically increasing over \(\mathbb{R}\) (since \( α≥0\)), while \(g\) is convex over \(\mathbb{R}^{n}\). Using the composition properties of the functions, we deduce that \(f\) is convex over \(\mathbb{R}^{n}\).

\(2.\) The function \(h(x)=e^{βx′Ax}\) can be viewed as a composition \(g(f(x))\) of the function \(g(t)=e^{βt}\) for \(t∈\mathbb{R}\) and the function \(f(x)=x′Ax\) for \(x∈\mathbb{R}^{n}\). In this case, \(g\) is convex and monotonically increasing over \(\mathbb{R}\), while \(f\) is convex over \(\mathbb{R}^{n}\) (since \(A\) is positive semidefinite), then \(f\) is convex over \(\mathbb{R}^{n}\).