To find the range of the function defined by the equation $y^2 = x^2 + 2x - 3$, we'll proceed as follows:

1. Rewrite the equation: $y^2 = x^2 + 2x - 3$
2. Complete the square for the expression on the right side to make it easier to analyze.

Completing the Square

Given: $y^2 = x^2 + 2x - 3$

Complete the square for the expression in $x$: $x^2 + 2x = (x + 1)^2 - 1$

Thus, the equation becomes: $y^2 = (x + 1)^2 - 1 - 3$ $y^2 = (x + 1)^2 - 4$

Analyze the Expression

We can rewrite this as: $y^2 + 4 = (x + 1)^2$

Since the left side represents $y^2 + 4$ and is always non-negative and greater than or equal to 4 (because $y^2 \geq 0$), the right side, $(x + 1)^2$, must also be greater than or equal to 4.

Thus, the expression $(x + 1)^2$ reaches its minimum value of 4 when $x + 1 = \pm 2$, i.e., $x = 1$ or $x = -3$.

Finding the Range of $y$

From the equation $y^2 + 4 = (x + 1)^2$, we can solve for $y^2$: $y^2 = (x + 1)^2 - 4$

Since $(x + 1)^2 \geq 4$, we have: $(x + 1)^2 - 4 \geq 0$

Thus: $y^2 \geq 0$

The smallest value $y^2$ can take is 0, which happens when $(x + 1)^2 = 4$. There is no upper bound for $y^2$, as $(x + 1)^2$ can increase indefinitely.

Therefore, the range of $y^2$ is $[0, \infty)$. Since $y^2 \geq 0$, the value of $y$ can be any real number, both positive and negative.

Range of the Function

The range of the function $y$ is: $y \in (-\infty, \infty)$

Do you have any questions, or would you like a more detailed explanation?

Here are some related questions you might explore:

1. What is the domain of the function $y^2 = x^2 + 2x - 3$?
2. How would the range change if the equation was $y = x^2 + 2x - 3$?
3. What is the range of the function $y = \sqrt{x^2 + 2x - 3}$?
4. How would the function change if we add a constant to both sides?
5. Can $y$ be a complex number in this equation?
6. What transformations are applied to the parent function $y = x^2$ in this case?
7. What is the significance of completing the square in this context?
8. How would the equation change if the term $-3$ was instead $+3$?

Tip: Completing the square is a valuable technique for transforming quadratic expressions, making it easier to analyze their properties such as vertex, range, and roots.