Find Domains: Analyzing y=-(x-3 1/12)² - 1/7 Function

The given quadratic function is $y = -\left(x - 3\frac{1}{12}\right)^2 - \frac{1}{7}$. We start by understanding that the shape of this parabola will open downwards due to the negative sign in front of the square term. To find the roots or x-intercepts, set $y = 0$.

Rewriting the expression for clarity, we have:

$0 = -\left(x - \frac{37}{12}\right)^2 - \frac{1}{7}$

We can solve this by isolating the squared term:

$\left(x - \frac{37}{12}\right)^2 = -\frac{1}{7}$

Since a squared term cannot be negative, it illustrates that there are no real roots. This means the parabola does not cross the x-axis and remains entirely below it, due to the downward opening.

Therefore, the function is negative ($y < 0$) for all x-values. The positive domain is non-existent.

The solution tells us:

• $x < 0 :$ all $x$
• $x > 0 :$ none

In terms of given choices: the correct choice is 3.

The solution to the problem is that the positive and negative domains are:

$x < 0 :$ all $x$

$x > 0 :$ none