To analyze the transformation from \( f(x) = 4x - 5 \) to \( 42x - 5 \), we first rewrite the original equation in a more familiar form:
- Original function: \( f(x) = 4x - 5 \)
- Transformed function: \( g(x) = 42x - 5 \)
The change from \( 4x \) to \( 42x \) indicates that the coefficient of \( x \) has increased from 4 to 42.
To relate this to the concept of stretch or compression:
- A function \( ax \) is a stretch if \( |a| > 1 \).
- A function \( ax \) is a compression if \( |a| < 1 \).
In this case, we are effectively multiplying the coefficient of \( x \) from 4 to 42.
To find the factor of transformation \( k \):
- Calculate \( k = \frac{42}{4} = 10.5 \).
Since \( k > 1 \), this indicates that there is a stretch.
Putting this together, the value of \( k \) would be \( 10.5 \), which is not provided in your options. However, if we assume the closest interpretation, focusing on the 2 that's present in the options, we see that none of them perfectly match the transformation we derived.
With respect to your options, none seem to be correct given the transformations. But the correct conceptual conclusion would be:
Conclusion:
- \( k = 10.5 \) indicating a stretch.
If the options only indicate a factor of increase by 2 rather than the actual numeric transformation, it could still indicate a stretch, but they don't accurately reflect the transformation in the original functions.
If we consider the problem at a basic level, the correct understanding of transformation suggests:
If \( f(x) \) moves from coefficient 4 to coefficient 42, that’s a significant stretch effect.
But if your context remains at the 2 factor as stated, the answer provided would seem incongruous with arithmetic. Please recheck your selections based on the clear transformation effects!