Graph the function f ( x )

Here's the graph of the function $f(x) = x + \frac{1}{(x - 3)(x - 1)}$ with the x-intercept and asymptotes displayed:

• x-intercept: $(-1, 0)$
• Vertical asymptotes: at $x = 1$ and $x = 3$
• Horizontal asymptote: $y = x$ as $x \to \pm \infty$

Do you have any questions about this graph, or would you like further details?

Here are some related questions for further exploration:

1. How do we determine the location of vertical asymptotes for rational functions?
2. What is the significance of horizontal asymptotes in rational functions?
3. How does changing the numerator affect the x-intercept of this function?
4. Can we predict the behavior near the asymptotes just by analyzing the function?
5. How would the graph change if the denominator was $(x - 3)^2(x - 1)$ instead?

Tip: When graphing rational functions, identifying the intercepts and asymptotes first helps to accurately sketch the overall shape.