It looks like you've uploaded an image of a graph representing a rational function. I'll analyze the graph to determine the rational function and solve it step by step.

Step 1: Identify Key Features of the Graph

Let's extract some information from the graph:

1. Vertical Asymptote: The graph shows a vertical line at $x = 0$. This indicates a vertical asymptote at $x = 0$, suggesting the denominator of the function has a factor of $x$.

2. Horizontal Asymptote: The graph levels off as $x$ increases in both directions, approaching a value close to $y = -1$. This suggests the horizontal asymptote is $y = -1$.

3. x-intercept: The graph crosses the x-axis at $x = 2$, so $f(2) = 0$.

4. y-intercept: There is no y-intercept since the function is undefined at $x = 0$.

Step 2: Formulate the Rational Function

Given these observations, the rational function likely has the form: $f(x) = \frac{a}{x} + b$ Where $a$ is a constant and $b$ is the horizontal asymptote (in this case, $b = -1$).

We know the x-intercept occurs at $x = 2$, so: $f(2) = \frac{a}{2} - 1 = 0$ Solving for $a$: $\frac{a}{2} = 1 \implies a = 2$

So, the rational function is: $f(x) = \frac{2}{x} - 1$

Step 3: Verify the Function

Let's verify that this function corresponds to the given graph:

• Vertical Asymptote: $x = 0$, matches the graph.
• Horizontal Asymptote: $y = -1$, matches the graph.
• x-intercept: $x = 2$, matches the graph.

Step 4: Conclusion

The rational function represented by the graph is $f(x) = \frac{2}{x} - 1$.

This matches the function you initially provided and confirms the analysis from earlier. Would you like to delve into any further details or discuss any particular aspect of this function?

Relative Questions

1. How can you determine the vertical asymptote by looking at the graph of a rational function?
2. What is the significance of the horizontal asymptote in understanding the end behavior of the function?
3. How would the function change if the horizontal asymptote were different (e.g., $y = 2$ instead of $y = -1$)?
4. What happens to the graph if the numerator in the function is altered (e.g., $\frac{3}{x} - 1$)?
5. How would the graph differ if the denominator was a polynomial of degree 2, like $\frac{2}{x^2} - 1$?

Tip

When identifying a rational function from its graph, pay close attention to the asymptotes and intercepts, as they provide crucial information about the function's structure.