## Functions Of Two Variables

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# Functions of Two Variables

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) =

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) =

3 - 2

f (2,0) =

0 - 2

Turning to another example, let us define z by

z

then for x = 3 and y = 4, we have z

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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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) =

__x__^{2}+ y^{2}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)__= 10,^{2}+ (3)^{2}3 - 2

f (2,0) =

__(2)__= -2,^{2}+ (0)^{2}0 - 2

Turning to another example, let us define z by

z

^{2}= x^{2}+ y^{2}then for x = 3 and y = 4, we have z

^{2}== 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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