Question

To transform f(x)=x^2+12
to f(x−3)
, which of the following shows the order in which operations should be performed?

a. square (x−3) , subtract 3, and add 12.
b. Subtract 3 from x, multiply by 2, and add 12.
c. Square (x−3), add 12, and subtract 3 from x.
d. Subtract 3 from x, square (x−3), and add 12.

Answer 1

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:

1. Substitute \( x - 3 \) in place of \( x \) in the function \( f(x) \):
\[
f(x - 3) = (x - 3)^2 + 12
\]

2. After that, we would compute \( (x - 3)^2 \) and then add 12.

Now, let's analyze the options provided:

- **a.** square (x−3), subtract 3, and add 12.
- This is incorrect because we don't subtract 3; we add 12 after squaring.

- **b.** Subtract 3 from x, multiply by 2, and add 12.
- This is incorrect because we don't multiply by 2.

- **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.

- **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:
1. Substitute \( x - 3 \) into \( f(x) \), which leads to \( f(x-3) = (x - 3)^2 + 12 \).
2. 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.