1. **State the problem:** We want to analyze the function $$y=\frac{1}{\sqrt{x^2+4x+3}}$$ and understand its behavior. 2. **Rewrite the expression inside the square root:** Factor the quadratic inside the square root: $$x^2+4x+3 = (x+1)(x+3)$$ 3. **Domain considerations:** Since the expression is under a square root in the denominator, the radicand must be positive (to avoid division by zero and complex numbers): $$x^2+4x+3 > 0$$ 4. **Solve the inequality:** The roots are at $$x=-3$$ and $$x=-1$$. The quadratic opens upward, so: $$x^2+4x+3 > 0 \implies x < -3 \text{ or } x > -1$$ 5. **Function behavior:** - For $$x < -3$$ or $$x > -1$$, the function is defined. - At $$x = -3$$ and $$x = -1$$, the denominator is zero, so vertical asymptotes occur. 6. **Final expression for graphing:** $$y=\frac{1}{\sqrt{(x+1)(x+3)}}$$ **Answer:** The function is defined for $$x < -3$$ and $$x > -1$$ with vertical asymptotes at $$x=-3$$ and $$x=-1$$.