Question
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What does the transformation f(x)↦f(x–1) do to the graph of f(x)?
Answers
GPT-4o mini
The transformation \( f(x) \mapsto f(x - 1) \) represents a horizontal shift of the graph of the function \( f(x) \). Specifically, this transformation shifts the graph of the function to the right by 1 unit.
To understand why this is the case, consider how the function is evaluated:
- For any input \( x \), the function \( f(x - 1) \) takes the value of \( f \) at \( x - 1 \).
- This means that to find the value of \( f(x - 1) \), you must look at the function's value one unit earlier on the x-axis.
As a result, if you were to take some point \( (a, f(a)) \) on the original graph \( y = f(x) \), after the transformation, this point would move to \( (a + 1, f(a)) \) on the transformed graph \( y = f(x - 1) \). Hence, every point on the graph moves to the right by 1 unit.
In summary, the transformation \( f(x) \mapsto f(x - 1) \) shifts the graph of \( f(x) \) to the right by 1 unit.
To understand why this is the case, consider how the function is evaluated:
- For any input \( x \), the function \( f(x - 1) \) takes the value of \( f \) at \( x - 1 \).
- This means that to find the value of \( f(x - 1) \), you must look at the function's value one unit earlier on the x-axis.
As a result, if you were to take some point \( (a, f(a)) \) on the original graph \( y = f(x) \), after the transformation, this point would move to \( (a + 1, f(a)) \) on the transformed graph \( y = f(x - 1) \). Hence, every point on the graph moves to the right by 1 unit.
In summary, the transformation \( f(x) \mapsto f(x - 1) \) shifts the graph of \( f(x) \) to the right by 1 unit.