1. **State the problem:** We need to sketch the graph of the function $$f(x) = \frac{x-1}{x+1}$$. 2. **Identify important features:** - The function is a rational function. - The denominator cannot be zero, so $$x \neq -1$$ (vertical asymptote). - To find horizontal asymptotes, analyze the behavior as $$x \to \pm \infty$$. 3. **Find vertical asymptote:** The denominator $$x+1=0$$ when $$x=-1$$, so there is a vertical asymptote at $$x=-1$$. 4. **Find horizontal asymptote:** For large $$x$$, $$f(x) \approx \frac{x}{x} = 1$$. So, horizontal asymptote is $$y=1$$. 5. **Find intercepts:** - **x-intercept:** Set numerator to zero: $$x-1=0 \Rightarrow x=1$$. - **y-intercept:** Set $$x=0$$: $$f(0) = \frac{0-1}{0+1} = -1$$. 6. **Analyze behavior near asymptotes:** - As $$x \to -1^-$$, denominator approaches 0 from negative side, numerator approaches $$-2$$, so $$f(x) \to +\infty$$. - As $$x \to -1^+$$, denominator approaches 0 from positive side, numerator approaches $$-2$$, so $$f(x) \to -\infty$$. 7. **Summary:** - Vertical asymptote at $$x=-1$$. - Horizontal asymptote at $$y=1$$. - x-intercept at $$x=1$$. - y-intercept at $$y=-1$$. This information allows sketching the graph accurately.