Let's analyze the function $g(x) = x^2$ to identify its key features such as domain, range, x-intercepts, y-intercepts, and intervals of increase and decrease:

1. Domain:

   • The domain of $g(x) = x^2$ is all real numbers, $\mathbb{R}$, because $x$ can take any real value.
   • Notation: $(-\infty, \infty)$

2. Range:

   • The range of $g(x) = x^2$ is all non-negative real numbers because the square of any real number is non-negative.
   • Notation: $[0, \infty)$

3. X-intercepts:

   • The x-intercept(s) occur where $g(x) = 0$. Solving $x^2 = 0$, we find $x = 0$.
   • Point: $(0, 0)$

4. Y-intercept:

   • The y-intercept occurs where $x = 0$. Substituting into the function, $g(0) = 0^2 = 0$.
   • Point: $(0, 0)$

5. Interval of Increase:

   • The function $g(x) = x^2$ increases on the interval where $x$ is positive, as $x$ gets larger, $x^2$ also gets larger.
   • Interval: $(0, \infty)$

6. Interval of Decrease:

   • The function decreases on the interval where $x$ is negative, as $x$ gets smaller (more negative), $x^2$ gets larger.
   • Interval: $(-\infty, 0)$

These are the key features of the quadratic function $g(x) = x^2$. If you need more details or have any questions, feel free to ask. Here are some related topics you might be interested in:

1. What is the vertex of the function $g(x) = x^2$?
2. How does the graph of $g(x) = x^2$ compare with $g(x) = -x^2$?
3. What would the graph look like if we add a constant term (e.g., $g(x) = x^2 + 3$)?
4. How do you find the axis of symmetry for the function $g(x) = x^2$?
5. Can you explain why the function has no real roots other than zero?

Tip: To understand the behavior of a quadratic function better, it's useful to remember that its graph is a parabola, which is symmetrical about the vertex.