Domain of Function: Finding Valid Inputs for (x+4 5/7)²

To determine the positive and negative domains of the function $y = \left(x + 4\frac{5}{7}\right)^2$, we need to analyze the behavior of the expression inside the square.

The expression $\left(x + 4\frac{5}{7}\right)^2$ represents a perfect square. A perfect square is always greater than or equal to zero.

Consider the following observations:

• The expression is zero when $x = -4\frac{5}{7}$. At this point, $y = 0$.
• For any other value of $x$, $x \neq -4\frac{5}{7}$, the expression $\left(x + 4\frac{5}{7}\right)^2$ is positive.

Given that $\left(x + 4\frac{5}{7}\right)^2$ cannot be negative for any real $x$, the function has no negative domain.

The positive domain of $y$ is all $x$ except when $x = -4\frac{5}{7}$. Hence, the positive domain is the set where $x \neq -4\frac{5}{7}$.

In conclusion, the negative domain is none, and the positive domain is $x \neq -4\frac{5}{7}$.

Therefore, the correct choice is:

$x < 0 :$ none

$x > 0 : x\ne-4\frac{5}{7}$