Question
The graph shown here is the graph of which of the following rational functions?
F(x) = 1/x+5
F(x) = 1/x-5
F(x) = 1/5x
F(x) = 1/x+5
F(x) = 1/x-5
F(x) = 1/5x
Answers
GPT-4o mini
To determine which rational function corresponds to the given graph, we need to look at the general characteristics of each function.
1. **F(x) = 1/(x + 5)**:
- This function has a vertical asymptote at \( x = -5 \).
- As \( x \) approaches -5 from the left, \( F(x) \) will approach negative infinity, and from the right, it will approach positive infinity.
- The horizontal asymptote is at \( y = 0 \).
2. **F(x) = 1/(x - 5)**:
- This function has a vertical asymptote at \( x = 5 \).
- As \( x \) approaches 5 from the left, \( F(x) \) will approach negative infinity, and from the right, it will approach positive infinity.
- The horizontal asymptote is at \( y = 0 \).
3. **F(x) = 1/(5x)**:
- This function has a vertical asymptote at \( x = 0 \).
- As \( x \) approaches 0 from the left, \( F(x) \) will approach negative infinity, and from the right, it will approach positive infinity.
- The horizontal asymptote is at \( y = 0 \).
Now, based on the vertical asymptotes, we can identify which function matches the graph:
- If the graph has a vertical asymptote at \( x = -5 \), then it corresponds to \( F(x) = 1/(x + 5) \).
- If the graph has a vertical asymptote at \( x = 5 \), then it corresponds to \( F(x) = 1/(x - 5) \).
- If the graph has a vertical asymptote at \( x = 0 \), then it corresponds to \( F(x) = 1/(5x) \).
By analyzing the vertical asymptote and behavior of the graph, you can select the correct function. Please refer to the graph for accurate observation.
1. **F(x) = 1/(x + 5)**:
- This function has a vertical asymptote at \( x = -5 \).
- As \( x \) approaches -5 from the left, \( F(x) \) will approach negative infinity, and from the right, it will approach positive infinity.
- The horizontal asymptote is at \( y = 0 \).
2. **F(x) = 1/(x - 5)**:
- This function has a vertical asymptote at \( x = 5 \).
- As \( x \) approaches 5 from the left, \( F(x) \) will approach negative infinity, and from the right, it will approach positive infinity.
- The horizontal asymptote is at \( y = 0 \).
3. **F(x) = 1/(5x)**:
- This function has a vertical asymptote at \( x = 0 \).
- As \( x \) approaches 0 from the left, \( F(x) \) will approach negative infinity, and from the right, it will approach positive infinity.
- The horizontal asymptote is at \( y = 0 \).
Now, based on the vertical asymptotes, we can identify which function matches the graph:
- If the graph has a vertical asymptote at \( x = -5 \), then it corresponds to \( F(x) = 1/(x + 5) \).
- If the graph has a vertical asymptote at \( x = 5 \), then it corresponds to \( F(x) = 1/(x - 5) \).
- If the graph has a vertical asymptote at \( x = 0 \), then it corresponds to \( F(x) = 1/(5x) \).
By analyzing the vertical asymptote and behavior of the graph, you can select the correct function. Please refer to the graph for accurate observation.