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.
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Understanding the original function: The function \( f(x) = x \) is a linear function with a slope of 1.
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Understanding the transformed function: The function \( g(x) = \frac{11}{12} x \) has a slope of \( \frac{11}{12} \).
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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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