1. The problem is to convert each given equation into function notation $f(x) = \ldots$ where $y$ is expressed as a function of $x$. 2. For $y = x$, the function notation is straightforward: $$f(x) = x$$ 3. For $3y = 2x$, solve for $y$: $$y = \frac{2x}{3}$$ So, $$f(x) = \frac{2x}{3}$$ 4. For $y + 3 = x$, solve for $y$: $$y = x - 3$$ So, $$f(x) = x - 3$$ 5. For $x + y = 0$, solve for $y$: $$y = -x$$ So, $$f(x) = -x$$ 6. For $y - x = 5$, solve for $y$: $$y = x + 5$$ So, $$f(x) = x + 5$$ 7. For $2y + x = 4y - 3x$, first simplify: $$2y + x = 4y - 3x$$ Bring all terms involving $y$ to one side and $x$ to the other: $$2y - 4y = -3x - x$$ $$-2y = -4x$$ Divide both sides by $-2$: $$y = 2x$$ So, $$f(x) = 2x$$ Final answers: $$f(x) = x, \quad f(x) = \frac{2x}{3}, \quad f(x) = x - 3, \quad f(x) = -x, \quad f(x) = x + 5, \quad f(x) = 2x$$