Domain Analysis: Finding Valid Inputs for y = (7/9)x² + 2

To solve this problem, we must determine when the quadratic function $y = \frac{7}{9}x^2 + 2$ is positive or negative.

• The function is a simple parabola opening upwards because the coefficient of $x^2$ is positive ($\frac{7}{9} > 0$). This indicates the graph of the function is positioned above the x-axis or tangent to it if it has roots (but here it clearly has no zero-crossings due to +2 as constant).

• Clearly, since this expression $\frac{7}{9}x^2$ is always non-negative for any real number $x$, plus 2, the value of $y$ is always positive.

• For $x < 0$, the term $x^2$ is still positive, hence $y$ remains positive. Therefore, there is no negative domain for $x < 0$.

• For $x > 0$, the quadratic behavior above justifies that $y$ is always positive across the entire positive domain of $x$.

By the analysis above:

$x < 0 : \text{none}$

$x > 0 : \text{all } x$

Thus, the correct choice is Choice 3:

$x < 0 : \text{none}$

$x > 0 : \text{all } x$

Therefore, the solution to the problem is $x < 0 :$ none, $x > 0 :$ all $x$.