Question
Select the correct answer. The graphs of rational functions \( m \) and \( n \) have the same vertical asymptotes, and both have a single \( x \)-interc Which equation could represent function \( n \) ? A. \( n(x)=5 m(x) \) B. \( n(x)=m(x)+5 \) C. \( n(x)=m(x-5) \) D. \( n(x)=m(x+5) \)
Ask by Fleming Perkins. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The correct equation for function \( n \) is \( n(x) = 5m(x) \).
Solution
To determine which equation could represent function \( n \), let's analyze the given information:
1. **Same Vertical Asymptotes**:
- Vertical asymptotes of a rational function are determined by the factors in the denominator. Since \( m \) and \( n \) have the same vertical asymptotes, their denominators must be identical. This means that any horizontal or vertical shifts would alter the position of these asymptotes.
2. **Single \( x \)-Intercept**:
- Both functions have exactly one \( x \)-intercept, meaning each function crosses the \( x \)-axis at only one point.
Now, let's evaluate each option:
- **A. \( n(x) = 5m(x) \)**
- **Effect**: This scales the function vertically by a factor of 5.
- **Impact on Asymptotes**: Vertical asymptotes remain unchanged because scaling does not affect the denominator.
- **Impact on \( x \)-Intercept**: The single \( x \)-intercept remains, as scaling does not change the location of the intercept.
- **B. \( n(x) = m(x) + 5 \)**
- **Effect**: This shifts the function vertically upward by 5 units.
- **Impact on Asymptotes**: Vertical asymptotes remain unchanged.
- **Impact on \( x \)-Intercept**: The \( x \)-intercept might no longer exist if the entire graph moves above the \( x \)-axis, or it might still intersect once depending on the original function.
- **C. \( n(x) = m(x - 5) \)**
- **Effect**: This shifts the function horizontally to the right by 5 units.
- **Impact on Asymptotes**: Vertical asymptotes would shift, which contradicts the given condition.
- **D. \( n(x) = m(x + 5) \)**
- **Effect**: This shifts the function horizontally to the left by 5 units.
- **Impact on Asymptotes**: Vertical asymptotes would shift, which contradicts the given condition.
**Conclusion:** Option **A** maintains the same vertical asymptotes and preserves the single \( x \)-intercept, making it the correct choice.
**Answer:**
**A. \(n(x)=5 m(x)\)**
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Simplify this solution Mind Expander
To maintain the same vertical asymptotes, the function \( n(x) \) should not change the denominator of \( m(x) \), which means we should avoid any transformations that would affect the locations of the vertical asymptotes. The correct answer is **B. \( n(x)=m(x)+5 \)** because this transformation shifts the entire function vertically but does not alter the vertical asymptotes that are defined by the denominator of \( m(x) \). On the other hand, options A, C, and D manipulate the function in ways that could potentially change the locations of the vertical asymptotes, making B the only viable choice to ensure they remain unchanged.
