1. **State the problem:** We are given the rational function $$y=\frac{2x+1}{x+3}$$ and want to understand its properties. 2. **Formula and rules:** A rational function is a ratio of two polynomials. Here, the numerator is $2x+1$ and the denominator is $x+3$. 3. **Domain:** The function is undefined where the denominator is zero, so solve $x+3=0$ which gives $x=-3$. Thus, the domain is all real numbers except $x=-3$. 4. **Simplify and analyze:** The function cannot be simplified further since numerator and denominator share no common factors. 5. **Find intercepts:** - **x-intercept:** Set numerator equal to zero: $2x+1=0 \Rightarrow x=-\frac{1}{2}$. - **y-intercept:** Set $x=0$: $$y=\frac{2(0)+1}{0+3}=\frac{1}{3}$$. 6. **Vertical asymptote:** At $x=-3$ because denominator is zero there. 7. **Horizontal asymptote:** Since degrees of numerator and denominator are equal (both degree 1), horizontal asymptote is ratio of leading coefficients: $$y=\frac{2}{1}=2$$. 8. **Summary:** - Domain: $x \neq -3$ - x-intercept: $x=-\frac{1}{2}$ - y-intercept: $y=\frac{1}{3}$ - Vertical asymptote: $x=-3$ - Horizontal asymptote: $y=2$