To determine the range of the function $f(x) = x^2$ for all real $x$:

Step-by-step Solution:

1. Definition of $f(x) = x^2$:

   • This is a quadratic function that outputs the square of any real number $x$.

2. Behavior of $x^2$:

   • For any real $x$, $x^2 \geq 0$.
   • Negative values for $f(x)$ are not possible because squaring a number always results in a non-negative value.

3. Minimum value:

   • The minimum value of $f(x)$ is $0$, which occurs when $x = 0$.

4. Maximum value:

   • There is no maximum value; as $x \to \infty$ or $x \to -\infty$, $f(x) \to \infty$.

5. Range:

   • The set of all possible values of $f(x)$ is $[0, \infty)$, which includes $0$ and all positive numbers.

Final Answer:

D. [0, ∞)

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Would you like further explanation or visualizations?
Here are some related questions for practice:

1. What is the domain of $f(x) = x^2$?
2. Find the range of $f(x) = (x - 3)^2 + 1$.
3. How does the range of $x^2$ change if the domain is restricted to $x \geq 0$?
4. What is the range of $f(x) = -x^2$?
5. Determine the range of $f(x) = x^2 + 2x + 1$.

Tip: To find the range of a function, analyze its behavior as $x \to \infty, x \to -\infty$, and its critical points.