The graph of a rational function yequals​f(x) is given. Use the graph to give the solution set of the following.
​(a)​ f(x)equals0
​(b) ​f(x)less than0
​(c)​ f(x)greater than0
Do not use a calculator.
-10
-5
5
10
-10
-5
5
10
x
y
y equals 1



A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A horizontal dashed line labeled y = 1 crosses the y-axis at 1. A graph has two branches. The first branch is below the horizontal dashed line and to the left of the y-axis, approaching both and passing through the point (negative 4, 0). The second branch is above the horizontal dashed line and to the right of the y-axis, approaching both.
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Part 1
​(a) What is the solution set of ​f(x)equals​0?

To solve this using the graph description:

### Part (a) - \( f(x) = 0 \)
The function \( f(x) = 0 \) corresponds to the points where the graph intersects the x-axis (where the y-value is 0). From the graph description, the first branch of the graph passes through the point \((-4, 0)\). Therefore, the solution to \( f(x) = 0 \) is:

**Solution:** \( x = -4 \)

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Would you like further clarification or a detailed explanation?

Here are some follow-up questions you might find useful:
1. What is a rational function, and how does it typically behave at vertical asymptotes?
2. What is the significance of the horizontal line \( y = 1 \) for this function?
3. Can the graph of a rational function cross its horizontal asymptote?
4. How would you find the vertical asymptote of a rational function from a graph?
5. What happens to the values of \( f(x) \) as \( x \to \infty \)?

**Tip:** When analyzing graphs, look for key points where the function touches or crosses the axes, which are often essential for finding roots or determining positive/negative intervals.

The graph of a rational function y = f(x) is given. Use the graph to give the solution set of the following. (a) f(x) = 0 (b) f(x) < 0 (c) f(x) > 0

Learn how to analyze a rational function graph to solve f(x) = 0, f(x) < 0, and f(x) > 0. This tutorial covers key concepts of rational functions, including finding solutions at x-intercepts and interpreting graph behavior near asymptotes. Suitable for high school students in Grades 9-12.

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