To determine which functions have the same transformations as \( f(x) = 2 \csc(x - 1) - 3 \), we should analyze the characteristics of each option regarding transformations.
The function \( f(x) = 2 \csc(x - 1) - 3 \) has the following transformations:
- A vertical stretch by a factor of 2.
- A horizontal shift to the right by 1 unit (due to \( x - 1 \)).
- A vertical shift downward by 3 units.
Now, let’s evaluate the options based on these transformations:
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\( f(x) = 2 (x - 1)^2 - 3 \):
- This is a quadratic function, and the transformations do not match the original function’s characteristics (sine and cosecant functions can be periodic). Thus, not a match.
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\( f(x) = \frac{2(x - 1)}{x - 3} \):
- This is a rational function and also does not match the characteristics of the original function. Thus, not a match.
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\( f(x) = 2 \sin(x - 3) \):
- This has a vertical stretch of 2 and a horizontal shift, but the horizontal shift is to the right by 3 units instead of 1 and it does not include the vertical shift of -3. Thus, not a match.
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\( f(x) = 2 e^{x - 1} - 3 \):
- Although there is a vertical stretch of 2 and a vertical shift of -3, this is an exponential function rather than a cosecant function and does not have the same periodicity or transformations. Thus, not a match.
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\( (x - 1)^2 + (y - 3)^2 = 2 \):
- This represents a circle centered at (1, 3) with a radius of \( \sqrt{2} \). This is a geometric representation and does not match the characteristics of the original function of cosecant, which is periodic and transformed differently. Thus, not a match.
None of the functions listed match the transformations of \( f(x) = 2 \csc(x - 1) - 3 \).
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