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Not all equations are equated to zero, but usually we set a function equal to zero if we want to find its x intercepts, or where the graph of the function crosses the x axis.
Set the first derivative of the function equal to zero, and solve for the variable.
The domain of a rational function is the whole of the real numbers except those points where the denominator of the rational function, simplified if possible, is zero.
The zero of a function is a point where the function evaluates to zero. If you express "y" as a function of "x", i.e. y = f(x), then for a zero of the function, the y-coordinate is 0. In other words, the corresponding point is on the x-axis.
You didn't specify the equation. A minimum or maximum value of a function is often found by calculating the derivative of a function, writing an equation for derivative equal to zero, and then analyzing points where the derivative either doesn't exist, or is equal to zero. You'll find find information about this in introductory calculus books.