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.