1. The problem asks for the range of the function $f(x) = x^2$. 2. The function $f(x) = x^2$ is a parabola opening upwards with vertex at $(0,0)$. 3. Since squaring any real number $x$ results in a value greater than or equal to zero, the smallest value of $f(x)$ is $0$. 4. As $x$ moves away from zero in either positive or negative direction, $f(x)$ increases without bound. 5. Therefore, the range of $f(x)$ is all real numbers $y$ such that $y \geq 0$. 6. In interval notation, this is $[0, \infty)$. 7. Comparing with the options, the correct answer is d. $[0, \infty)$.