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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.

Identify the key features of the function g(x) = x^2, including the domain, range, x and y intercepts, and intervals of increase and decrease.

Explore the key features of the quadratic function g(x) = x^2, including its domain, range, x and y intercepts, and intervals of increase and decrease. This problem is ideal for students in grades 9-10 learning about quadratic functions.

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