1. **State the problem:** We are given the function $$f(x) = \frac{x^{2}}{\sqrt{x^{2} + 2x - 3}}$$ and need to analyze it, including its domain and behavior. 2. **Identify the domain:** The denominator contains a square root, so the expression inside must be positive (not zero or negative) to avoid division by zero or imaginary numbers. 3. **Set the radicand greater than zero:** $$x^{2} + 2x - 3 > 0$$ 4. **Factor the quadratic:** $$x^{2} + 2x - 3 = (x + 3)(x - 1)$$ 5. **Solve the inequality:** $$(x + 3)(x - 1) > 0$$ This product is positive when both factors are positive or both are negative. - Both positive: $$x - 1 > 0 \Rightarrow x > 1$$ - Both negative: $$x + 3 < 0 \Rightarrow x < -3$$ So the domain is: $$(-\infty, -3) \cup (1, \infty)$$ 6. **Check the numerator:** The numerator is $$x^{2}$$ which is always non-negative. 7. **Behavior near domain boundaries:** At $$x = -3$$ and $$x = 1$$, the denominator approaches zero, so the function tends to infinity or negative infinity depending on the side. 8. **Summary:** - Domain: $$(-\infty, -3) \cup (1, \infty)$$ - Function is undefined between $$-3$$ and $$1$$. 9. **Desmos function for graphing:** $$y = \frac{x^{2}}{\sqrt{x^{2} + 2x - 3}}$$ This matches the given function and shows the curve with vertical asymptotes at $$x = -3$$ and $$x = 1$$. **Final answer:** The domain of $$f(x)$$ is $$(-\infty, -3) \cup (1, \infty)$$.