To determine the transformation that takes [tex]\( f(x) = x^2 - 1 \)[/tex] to [tex]\( g(x) = (x-1)^2 - 1 \)[/tex], we need to analyze how the form of the function changes.
1. Start with the original function:
[tex]\[
f(x) = x^2 - 1
\][/tex]
2. Now, consider the transformed function:
[tex]\[
g(x) = (x-1)^2 - 1
\][/tex]
3. Notice that the term [tex]\((x-1)\)[/tex] inside the square in [tex]\(g(x)\)[/tex] indicates a horizontal shift. To identify the direction of this shift, recall that:
- [tex]\( f(x+h) \)[/tex] represents a horizontal translation [tex]\( h \)[/tex] units to the left.
- [tex]\( f(x-h) \)[/tex] represents a horizontal translation [tex]\( h \)[/tex] units to the right.
4. The expression [tex]\((x-1)\)[/tex] corresponds to [tex]\( x \)[/tex] having 1 subtracted from it, which matches the format [tex]\( x-h \)[/tex].
[tex]\[
(x-1) \Rightarrow x - 1
\][/tex]
5. Therefore, the transformation [tex]\( g(x) = (x-1)^2 - 1 \)[/tex] implies a horizontal shift 1 unit to the right of the [tex]\( x \)[/tex]-value in the argument of the original function [tex]\( f(x) \)[/tex].
6. The rest of the function, [tex]\(- 1\)[/tex], remains unchanged, indicating no vertical shift occurs.
Thus, the correct answer that describes the transformation of [tex]\( f(x) = x^2 - 1 \)[/tex] to [tex]\( g(x) = (x-1)^2 - 1 \)[/tex] is:
[tex]\[
\text{a horizontal translation 1 unit to the right}
\][/tex]
