1. **State the problem:** We need to analyze the features of the graph of the function $$f(x) = \frac{x^2 - 1}{x - 1}$$ including intercepts, asymptotes, and holes. 2. **Simplify the function:** Factor the numerator: $$x^2 - 1 = (x - 1)(x + 1)$$ So, $$f(x) = \frac{(x - 1)(x + 1)}{x - 1}$$ 3. **Cancel common factors:** Since $$x - 1$$ appears in numerator and denominator, cancel it out, but note that $$x \neq 1$$ because it makes denominator zero: $$f(x) = \cancel{\frac{(x - 1)(x + 1)}{x - 1}} = x + 1, \quad x \neq 1$$ 4. **Identify holes:** The factor $$x - 1$$ canceled means there is a hole at $$x = 1$$. Find the y-coordinate of the hole by substituting $$x = 1$$ into the simplified function: $$f(1) = 1 + 1 = 2$$ So, hole at $$(1, 2)$$. 5. **Vertical asymptotes:** Since the factor causing zero denominator was canceled, there is no vertical asymptote. 6. **Horizontal asymptotes:** The simplified function is linear $$y = x + 1$$, which has no horizontal asymptote. 7. **Intercepts:** - **x-intercept:** Set $$f(x) = 0$$: $$x + 1 = 0 \Rightarrow x = -1$$ So x-intercept is $$(-1, 0)$$. - **y-intercept:** Evaluate $$f(0)$$: $$f(0) = 0 + 1 = 1$$ So y-intercept is $$(0, 1)$$. **Final summary:** - Hole at $$(1, 2)$$ - No vertical asymptote - No horizontal asymptote - x-intercept at $$(-1, 0)$$ - y-intercept at $$(0, 1)$$