To determine which function has been horizontally stretched by a factor of 74 and reflected across the y-axis, we need to understand the transformations involved.
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Horizontal Stretch: A horizontal stretch by a factor of \( k \) can be represented by replacing \( x \) with \( \frac{x}{k} \). For a stretch by a factor of 74, we replace \( x \) with \( \frac{x}{74} \).
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Reflection across the y-axis: This transformation affects the sign of \( x \) in the function, so we replace \( x \) with \( -x \).
Combining these two transformations, starting from the function \( f(x) = |x| \):
- Apply the reflection: \( f(-x) = |-x| = |x| \) (the absolute value remains unchanged).
- Apply the horizontal stretch: \( f\left(-\frac{x}{74}\right) = \left| -\frac{x}{74} \right| = \frac{|x|}{74} \).
Thus, if we incorporate both transformations together, we start with \( |x| \) and apply the horizontal stretch followed by the reflection:
\[ f(x) = \left| -\frac{x}{74} \right| = \frac{|x|}{74} \]
This function indicates a transformation of \( f(x) \) to \( |-\frac{x}{74}| \).
Now let's look at the provided options:
- \( f(x) = |47x| \) - Not correct.
- \( f(x) = |\frac{4}{7} x| \) - Not correct.
- \( f(x) = |74x| \) - Not correct.
- \( f(x) = |-74x| \) - This represents reflection and stretching combined correctly.
Hence, the correct function that has been horizontally stretched by a factor of 74 and reflected across the y-axis is:
\( f(x) = |-74x| \).