It seems like you meant to write \( f(x) = x^2 \). This represents a quadratic function, where the output \( f(x) \) is equal to the square of the input \( x \).

### Key Characteristics of \( f(x) = x^2 \):

1. **Domain**: The domain is all real numbers, \( x \in \mathbb{R} \), because you can square any real number.
2. **Range**: The range is \( [0, \infty) \), since squaring any real number yields a non-negative result.
3. **Vertex**: The vertex of the parabola represented by this function is at the origin (0, 0).
4. **Axis of Symmetry**: The axis of symmetry is the vertical line \( x = 0 \).
5. **Intercepts**:
   - **x-intercept**: The function intersects the x-axis at \( (0, 0) \).
   - **y-intercept**: The function intersects the y-axis at \( (0, 0) \).
6. **Behavior**:
   - The function is increasing when \( x > 0 \).
   - The function is decreasing when \( x < 0 \).
7. **Graph Shape**: The graph is a parabola opening upwards.
8. **Concavity**: The function is concave up for all \( x \).

Would you like more details on any aspect, or do you have any other questions?

### Related Questions:
1. What is the effect of adding a constant term to the function \( f(x) = x^2 \)?
2. How does the graph of \( f(x) = -x^2 \) differ from \( f(x) = x^2 \)?
3. What happens if you add a linear term, such as \( f(x) = x^2 + 2x \)?
4. How do you find the vertex of a general quadratic function \( f(x) = ax^2 + bx + c \)?
5. What is the significance of the discriminant in a quadratic equation?
6. How do you complete the square for the function \( f(x) = x^2 + 6x + 9 \)?
7. How can you solve the equation \( x^2 = 25 \)?
8. What are the properties of the function \( f(x) = \sqrt{x} \) compared to \( f(x) = x^2 \)?

**Tip:** When sketching the graph of a quadratic function, always start by identifying the vertex and the direction in which the parabola opens.

Explore the quadratic function f(x) = x^2 in detail, covering its domain, range, vertex, axis of symmetry, intercepts, behavior, and graph shape. This comprehensive guide provides insights into quadratic functions, suitable for Grades 9-12.

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