A2 The set of real numbers $k$ for which the function $f(x)=x^{2}-(k-1) x+1$ is positive for every $x \in \mathbb{R}$, is:
(A) $(-\infty,-1)$
(B) $(-2,2)$
(C) $[-1,3)$
(D) $(-1,3)$
(E) $(-1,3]$
A2. The quadratic function is positive if $D<0$. It must be true that $(k-1)^{2}-4 \cdot 1 \cdot 1<0$, thus $k^{2}-2 k-3<0$. Solve the inequality by finding the solutions to the equation $k^{2}-2 k-3=0$, factorize $(k-3)(k+1)=0$, and read off $k_{1}=-1$ and $k_{2}=3$. Check the signs on the number line and get the solution D.