To find the domain and range of the function \( y = x^2 - 3 \), let's analyze the function step by step.

### Domain:
The domain of a function consists of all the possible input values (x-values) that the function can accept. For the function \( y = x^2 - 3 \):

- The function involves a polynomial expression, \( x^2 \), which is defined for all real numbers.
- There are no restrictions like division by zero or square roots of negative numbers.

**Domain**: The domain is all real numbers, which can be written as:
\[
\text{Domain} = (-\infty, \infty)
\]

### Range:
The range of a function consists of all the possible output values (y-values) that the function can produce.

- The expression \( x^2 \) is always non-negative (i.e., \( x^2 \geq 0 \)).
- The smallest value of \( x^2 \) is 0, which occurs when \( x = 0 \). So, the smallest value of \( y \) is:
\[
y = 0^2 - 3 = -3
\]
- As \( x^2 \) increases (as \( x \) moves away from 0 in either direction), \( y \) increases without bound.

**Range**: The range is all real numbers greater than or equal to \(-3\), which can be written as:
\[
\text{Range} = [-3, \infty)
\]

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How would the domain and range change if the function was \( y = \sqrt{x^2 - 3} \)?
2. What is the vertex of the parabola represented by the function \( y = x^2 - 3 \)?
3. How does shifting the function vertically affect the range?
4. What would the domain and range be for the function \( y = x^2 + 2x - 3 \)?
5. How can we determine the axis of symmetry for the function \( y = x^2 - 3 \)?

**Tip**: For any quadratic function in the form \( y = ax^2 + bx + c \), the vertex is the point that gives the minimum or maximum value of the function, which is useful in determining the range.

Discover how to find the domain and range of the quadratic function y = x^2 - 3. This lesson explains the core concepts of algebra and quadratic functions, perfect for students in Grades 8-10.

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