1. The problem asks to classify each of the five expressions as functions or not functions. 2. Recall that a function assigns exactly one output for each input. 3. Expression 1: $y = [[x]]$ where $[[x]]$ is the greatest integer function (floor function). This is a function because for each $x$, $[[x]]$ gives a unique integer. 4. Expression 2: $y = [[x]] - xy$. Here, $y$ appears on both sides, so to check if this defines a function, solve for $y$: $$y = [[x]] - x y$$ $$y + x y = [[x]]$$ $$y(1 + x) = [[x]]$$ $$y = \frac{[[x]]}{1 + x}$$ Since $1 + x \neq 0$ for $x \neq -1$, this defines a unique $y$ for each $x \neq -1$. At $x = -1$, denominator is zero, so no function value. So this is a function on domain $x \neq -1$. 5. Expression 3: $y^2 = [[x]] + x y^2$. Rearranged: $$y^2 - x y^2 = [[x]]$$ $$y^2(1 - x) = [[x]]$$ $$y^2 = \frac{[[x]]}{1 - x}$$ For each $x$ where denominator is not zero, $y^2$ equals some value. But $y$ is squared, so for positive right side, $y$ can be $\pm \sqrt{...}$, two values. So this is not a function because one input $x$ can give two $y$ values. 6. Expression 4: $x^2 + y^2 = 4$. This is a circle of radius 2. For many $x$ values, there are two $y$ values (positive and negative). So not a function. 7. Expression 5: $y = x^2$. This is a parabola and a function because for each $x$ there is exactly one $y$. Final classification: 1) yes 2) yes 3) no 4) no 5) yes