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To solve for the equation of \( f(x) \), let’s analyze the information given:

1. **Vertical Asymptotes**: The function has vertical asymptotes at \( x = -1 \) and \( x = 3 \). This suggests that the denominator of \( f(x) \) should include factors \( (x + 1) \) and \( (x - 3) \).
   
2. **Horizontal Asymptote**: The function has a horizontal asymptote at \( y = 0 \). For a rational function to have a horizontal asymptote at zero, the degree of the numerator must be less than the degree of the denominator. This implies that the numerator should be a constant or a first-degree polynomial.

3. **X-Intercept**: The function has an \( x \)-intercept at \( x = 1 \). For a rational function to have an \( x \)-intercept at \( x = 1 \), the numerator should include \( (x - 1) \) as a factor.

4. **Point (2,1)**: The function passes through the point \( (2, 1) \), which we can use to solve for any constant in the equation.

### Choosing the Form of \( f(x) \)

Given the information, a suitable form for \( f(x) \) would be:
\[
f(x) = \frac{a(x - 1)}{(x + 1)(x - 3)}
\]
since this form accommodates two vertical asymptotes and an \( x \)-intercept.

### Finding the Value of \( a \)

To determine \( a \), we use the point \( (2, 1) \):
\[
f(2) = 1 \Rightarrow \frac{a(2 - 1)}{(2 + 1)(2 - 3)} = 1
\]
Simplifying:
\[
\frac{a \cdot 1}{3 \cdot -1} = 1 \Rightarrow \frac{-a}{3} = 1
\]
\[
a = -3
\]

### Final Equation

The equation for \( f(x) \) is:
\[
f(x) = \frac{-3(x - 1)}{(x + 1)(x - 3)}
\]

Let me know if you’d like further explanation or have questions.

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Here are five related questions to consider:

1. How would the equation change if there were an additional vertical asymptote at another \( x \)-value?
2. What would the function look like if it had a different horizontal asymptote, such as \( y = 2 \)?
3. How does the degree of the polynomial in the numerator affect the horizontal asymptote?
4. Why does the factor \( (x - 1) \) in the numerator result in an \( x \)-intercept at \( x = 1 \)?
5. How can you verify that the function passes through a specific point?

**Tip:** When determining a rational function, identify vertical asymptotes from the denominator and \( x \)-intercepts from the numerator factors.

The figure below shows the graph of a rational function f. It has vertical asymptotes x = -1 and x = 3, and horizontal asymptote y = 0. The graph has x-intercept 1, and it passes through the point (2, 1). The equation for f(x) has one of the five forms shown below. Choose the appropriate form for f(x), and then write the equation. You can assume that f(x) is in simplest form.

Learn how to find the equation of a rational function given its vertical asymptotes, horizontal asymptote, x-intercept, and a point through which it passes. This problem explores the properties of rational functions and asymptotes, suitable for students in Grades 11-12.

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