Find the Domain of y=-x²-4⅓: Complete Function Analysis

The quadratic function is $y = -x^2 - \frac{4}{3}$. This function graphs as a parabola opening downwards.

Let's analyze the sign of the function:

• Since the parabola opens downwards (negative coefficient of $x^2$), $y$ takes on non-positive values. No solution exists for when $y > 0$.
• For $y = 0$, solving gives: $-x^2 - \frac{4}{3} = 0$ which means $-x^2 = \frac{4}{3}$. This equation cannot be true for any real $x$ because squares are non-negative and cannot yield negative values.
• Therefore, the function does not cross or even touch the $x$-axis.
• This means the range of the function simply yields negative values for any $x$.

Let's review some regions:

• Negative Domain, $x < 0$: Since the function always remains negative or zero, for negative $x$, $\,$ the domain technically spans all real numbers, but function values will be less than 0 (negative).
• Positive Domain, $x > 0$: None as $y > 0$ does not happen for any real $x$ due to the nature of the parabola being downward opening and completely below the $x$-axis.

The correct choice identifying domains is: $x < 0 :$ all $x$

$x > 0 :$ none.