To determine the location of the point after a horizontal translation of the original function \( y = f(x) \) that goes through the point (1, 1), we need to consider the nature of horizontal translations.
- A horizontal translation to the right by \( h \) units transforms \( f(x) \) into \( f(x - h) \).
- A horizontal translation to the left by \( h \) units transforms \( f(x) \) into \( f(x + h) \).
In your question, it appears that the point (1, 1) initially lies on the curve defined by \( y = f(x) \). Without knowing specifically how many units the translation has occurred (to the left or right), we apply the translation results to determine the new point and function.
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If translated to the right (for example \( h = 2 \)):
- Original point: \( (1, 1) \)
- Translated point: \( (1 + 2, 1) = (3, 1) \)
- Translated function: \( f(x - 2) \)
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If translated to the left (for example \( h = 2 \)):
- Original point: \( (1, 1) \)
- Translated point: \( (1 - 2, 1) = (-1, 1) \)
- Translated function: \( f(x + 2) \)
Given the responses you provided, it seems you're looking for translations specifically either to the right or left, likely by 2 units based on the options showing 2 as the distance.
The correct match from your responses, through reasoning, is that the point ends up at \( (3, 1) \) if translated to the right by 2 units, leading us to choose:
- The point is located at \( (3, 1) \), and the translated function is \( y = f(x - 2) \).
- The point is located at \textbf{(3, 1)}, and the translated function is \( f(x - 2) \).
This is repeated in both last options. Therefore, either of these statements corresponds to the condition of having the point after translation, leading us to conclude:
The point is located at \( (3, 1) \), and the translated function is \( y = f(x - 2) \).
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