The question asks for the range of the function $f(x) = x^2$.

Solution:

The function $f(x) = x^2$ is a quadratic function and is a parabola that opens upwards.

1. Domain: The domain of $f(x) = x^2$ is all real numbers ($x \in \mathbb{R}$) because you can plug any real number into $x^2$.

2. Range: The range of $f(x) = x^2$ consists of all the possible output values. Since squaring any real number results in a non-negative number, the output (range) will always be $y \geq 0$.

   • The smallest value of $f(x)$ occurs at $x = 0$, where $f(0) = 0^2 = 0$.
   • As $x$ moves away from 0 in either the positive or negative direction, $f(x)$ increases.

Therefore, the range of the function $f(x) = x^2$ is all real numbers greater than or equal to 0, which is:

$\boxed{[0, \infty)}$

Correct Answer:

Looking at the provided multiple-choice options, the correct answer is:

• $\textbf{D) f(x) \geq 0}$

If you have any questions or need further explanation, feel free to ask!

Related Questions:

1. What is the domain of the function $f(x) = x^2$?
2. How does the graph of $y = x^2$ compare to the graph of $y = -x^2$?
3. How do you find the vertex of a parabola in general form?
4. What is the effect of adding a constant to the function $f(x) = x^2$?
5. How would the range change for the function $g(x) = (x-3)^2 + 1$?

Tip:

When analyzing the range of a function, consider the possible output values that the function can produce based on its mathematical expression and any restrictions it might have.