If to each point (x.y) in a certain subset of the two-dimensional plane there corresponds one and only value of z, then the correspondence is called a function of two variables. Just as in the one independent variable case, this correspondence is often represented by an equation. However, the equation is now of the form

z = f (x,y),

where z is the dependent variable, and x and y are the independent variables. The set of all allowable values for the independent variables constitutes the domain of the function. For example, the equation.

z = f (x,y) = x² + y²
                    y - 2

defines z as a function of x and y. Because the denominator is zero when y =2 the domain of f is all ordered pairs (x,y) such that y ≠ 2. Some function values are

f (1,3) = (1)² + (3)²  = 10,
              3 - 2

f (2,0) = (2)² + (0)²  = -2,
              0 - 2

Turning to another example, let us define z by

z² = x² + y²

then for x = 3 and y = 4, we have z² == 25.

Consequently, z = ±  5. Thus with the ordered pair (3,4) w cannot associate exactly one value of z . Hence z is not a function of x and y.

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