Asked by goku dragon and naruto dragon
What is the effect on the graph of f(x)=1x when it is replaced by g(x)=1112x ? Option #1: vertically compressed by a factor of 112 Option #2: horizontally stretched by a factor of 12 Option #3: horizontally compressed by a factor of 112 Option #4: horizontally shifted left by 112 unit
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Answered by
GPT-4o mini
To analyze the effect of replacing \( f(x) = x \) with \( g(x) = \frac{11}{12} x \), we need to consider how the transformation impacts the graph of the function.
1. **Understanding the original function**: The function \( f(x) = x \) is a linear function with a slope of 1.
2. **Understanding the transformed function**: The function \( g(x) = \frac{11}{12} x \) has a slope of \( \frac{11}{12} \).
3. **Effect on the graph**:
- The slope \( \frac{11}{12} \) is less than 1, which means that the line will be less steep compared to \( f(x) = x \).
- This can be interpreted as a vertical compression because the output values will be smaller compared to the original function's output for positive \( x \), and larger negative for negative \( x \).
**Conclusion**: The correct interpretation of the transformation from \( f(x) \) to \( g(x) \) indicates that the graph has been vertically compressed by a factor of \( \frac{12}{11} \) because we are multiplying by a fraction less than 1.
In terms of the options provided:
- Option #1: "vertically compressed by a factor of \( \frac{1}{12} \)" is somewhat misleading because it doesn't state accurately how the transformation occurs.
However, if we interpret "vertically compressed by a factor of \( \frac{12}{11} \)" to mean that the output values are reduced compared to \( f(x) \), we can say **Option #1 is the closest** to being correct regarding vertical compression, but the fraction should have been \( \frac{12}{11} \) for accuracy.
Based on the options available:
**Option #1: vertically compressed by a factor of 12** (assuming this is a typographical error and refers to \( \frac{12}{11} \)).
1. **Understanding the original function**: The function \( f(x) = x \) is a linear function with a slope of 1.
2. **Understanding the transformed function**: The function \( g(x) = \frac{11}{12} x \) has a slope of \( \frac{11}{12} \).
3. **Effect on the graph**:
- The slope \( \frac{11}{12} \) is less than 1, which means that the line will be less steep compared to \( f(x) = x \).
- This can be interpreted as a vertical compression because the output values will be smaller compared to the original function's output for positive \( x \), and larger negative for negative \( x \).
**Conclusion**: The correct interpretation of the transformation from \( f(x) \) to \( g(x) \) indicates that the graph has been vertically compressed by a factor of \( \frac{12}{11} \) because we are multiplying by a fraction less than 1.
In terms of the options provided:
- Option #1: "vertically compressed by a factor of \( \frac{1}{12} \)" is somewhat misleading because it doesn't state accurately how the transformation occurs.
However, if we interpret "vertically compressed by a factor of \( \frac{12}{11} \)" to mean that the output values are reduced compared to \( f(x) \), we can say **Option #1 is the closest** to being correct regarding vertical compression, but the fraction should have been \( \frac{12}{11} \) for accuracy.
Based on the options available:
**Option #1: vertically compressed by a factor of 12** (assuming this is a typographical error and refers to \( \frac{12}{11} \)).
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