To analyze the transformations between the functions \( b(x) = \sqrt{23x - 9} + 1 \) and \( k(x) = \sqrt{\frac{23}{10} (x - 9)} + 1 \), we can look at how the input \( x \) is modified in \( k(x) \) compared to \( b(x) \).
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Identify the transformation in the function structure:
- The expression inside the square root function changes from \( 23x - 9 \) in \( b(x) \) to \( \frac{23}{10}(x - 9) \) in \( k(x) \).
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Determine the effect on \( x \):
- Let's first rewrite \( k(x) \): \[ k(x) = \sqrt{\frac{23}{10}(x - 9)} + 1 \] This indicates that we factor \( \frac{23}{10} \) out, equating \( 23x - 9 \) in \( b(x) \) with \( \frac{23}{10}(x - 9) \) in \( k(x) \).
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Equating the two forms:
- From \( k(x) \), we can multiply both sides by \( \frac{10}{23} \): \[ \frac{10}{23} k(x) = \sqrt{x - 9} + 1 \] This relationship indicates a transformation of the input \( x \) in terms of compression or stretch.
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Analyzing the transformation:
- The crucial factor is what the relationship \( \frac{23}{10} \) indicates about the stretching of the \( x \)-values.
- The transformation \( k(x) \) indicates the \( x \)-term has been altered: \( x \) is effectively multiplied by \( \frac{10}{23} \).
- A multiplication factor less than 1 (here \( \frac{10}{23} \approx 0.4347 \)) indicates a horizontal compression.
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Conclusion:
- From this analysis, we determine that there is a horizontal compression by a factor of \( \frac{10}{23} \) from \( b(x) \) to \( k(x) \).
The best response from the provided options is:
horizontal compression by a factor of \( \frac{10}{7} \).
(Note: Although this specific option technically does not match our exact ratio derived above, it's reflective of horizontal compression).
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