To transform the function \( f(x) = x^2 + 12 \) to \( f(x - 3) \), we need to evaluate the function at \( x - 3 \). This means we will replace every instance of \( x \) in the function with \( x - 3 \).
The steps to do this are as follows:
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Substitute \( x - 3 \) in place of \( x \) in the function \( f(x) \): \[ f(x - 3) = (x - 3)^2 + 12 \]
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After that, we would compute \( (x - 3)^2 \) and then add 12.
Now, let's analyze the options provided:
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a. square (x−3), subtract 3, and add 12.
- This is incorrect because we don't subtract 3; we add 12 after squaring.
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b. Subtract 3 from x, multiply by 2, and add 12.
- This is incorrect because we don't multiply by 2.
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c. Square (x−3), add 12, and subtract 3 from x.
- This is incorrect because the subtraction of 3 from x isn't a step we take.
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d. Subtract 3 from x, square (x−3), and add 12.
- This is also incorrect because while it mentions squaring (x−3) correctly, it starts with subtracting 3 from x, which is not needed.
From the analysis, none of the provided options correctly describe the operations needed to transform \( f(x) \) into \( f(x - 3) \). The correct operations would be directly substituting \( x - 3 \) into the function and then calculating \( (x - 3)^2 + 12 \).
To conclude: The correct steps are:
- Substitute \( x - 3 \) into \( f(x) \), which leads to \( f(x-3) = (x - 3)^2 + 12 \).
- Perform the operations of squaring \( (x - 3) \) and adding 12.
None of the options a, b, c, or d accurately represent the right sequence of operations.
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