Domain Analysis of y=-1/6x²-5/7: Finding Valid Input Values

To solve the problem, we proceed as follows:

• The quadratic function is given as $y = -\frac{1}{6}x^2 - \frac{5}{7}$.
• The coefficient $a = -\frac{1}{6}$ is negative, indicating the parabola opens downward.
• The function has no $x$-term ($b = 0$), so the vertex $x$-coordinate is $0$.
• Evaluating $y$ at $x = 0$, we have: $y = -\frac{5}{7}$, which is negative.
• Since the parabola opens downward and $y$ at the vertex is negative, $y$ remains negative for all $x$ values.
• Thus, there is no positive domain.
• For $x < 0$, $y$ is negative for all $x$.

In conclusion, the function remains negative for all $x < 0$ and is also negative for all else. The answer choice matches:

$x < 0 :$ all $x$

$x > 0 :$ none