1. **State the problem:** We are given the function $f(x) = \frac{2}{x+1}$ and we want to understand its behavior. 2. **Formula and rules:** This is a rational function where the numerator is constant and the denominator is a linear expression. Important rules: - The function is undefined where the denominator is zero. - The domain excludes values that make the denominator zero. 3. **Find the domain:** Set denominator equal to zero: $$x+1=0$$ $$x=-1$$ So, $x=-1$ is not in the domain. 4. **Simplify and analyze:** The function cannot be simplified further. It has a vertical asymptote at $x=-1$. 5. **Horizontal asymptote:** As $x \to \pm \infty$, $f(x) \to 0$ because numerator is constant and denominator grows large. 6. **Summary:** - Domain: $x \neq -1$ - Vertical asymptote: $x=-1$ - Horizontal asymptote: $y=0$ This function decreases on intervals $(-\infty,-1)$ and $(-1,\infty)$ and is positive for all $x \neq -1$.