Mathematics

For $f_k(x)$ to be defined for all real $x$, the expression under the square root must never be negative. This means the quadratic $x^2 + 2kx + 9$ must be non-negative for every $x \in \mathbb{R}$—in other words, it can't dip below the $x$-axis. A quadratic stays non-negative everywhere exactly when its discriminant is non-positive, so you need to find which values of $k$ make the discriminant $\leq 0$.

For $f_k(x)$ to be defined for all real $x$, the expression under the square root must never be negative. This means the quadratic $x^2 + 2kx + 9$ must be non-negative for every $x \in \mathbb{R}$—in other words, it can't dip below the $x$-axis. A quadratic stays non-negative everywhere exactly when its discriminant is non-positive, so you need to find which values of $k$ make the discriminant $\leq 0$.