Find the Domain of (x-8)²-1: Complete Function Analysis

To solve for the positive and negative domains of $y = (x-8)^2 - 1$:

• Identify when $y$ equals zero. Set the equation $(x-8)^2 - 1 = 0$.
• Add 1 to both sides: $(x-8)^2 = 1$.
• Take the square root: $x-8 = \pm 1$.
• Solve for $x$: $x = 8 \pm 1 \rightarrow x = 7$ or $x = 9$.

So, the function $y = (x-8)^2 - 1$ is zero at $x = 7$ and $x = 9$. These points divide the x-axis into intervals:

• Interval 1: $x < 7$, test $x = 6$. $(6-8)^2 - 1 = 3$, which is positive.
• Interval 2: $7 < x < 9$, test $x = 8$. $(8-8)^2 - 1 = -1$, which is negative.
• Interval 3: $x > 9$, test $x = 10$. $(10-8)^2 - 1 = 3$, which is positive.

From this analysis:

The negative domain (where $y < 0$) is $7 < x < 9$.

The positive domain (where $y > 0$) consists of $x < 7$ and $x > 9$.

Therefore, the correct answer from the provided choices is:

x < 0 : 7 < x < 9

x > 9 or x > 0 : x < 7