Math 22 Functions of Several Variables

Definition of a Function of Two Variables

 Let $D$ be a set of ordered pairs of real numbers. 
 If to each ordered pair $(x,y)$ in $D$ there corresponds a unique real number $f(x,y)$, then $f$ is a function of $x$ and $y$. 
 The set $D$ is the domain of $f$, and the corresponding set of values for $f(x,y)$ is the range of $f$. Functions of three, four, or more variables are defined similarly.

Exercises 1 Given $f(x,y)=2x+y-3$. Evaluate:

1) $f(0,2)$

Solution:
$f(x,y)=2x+y-3$
So, $f(0,2)=2(0)+2-3=-1$

2) $f(5,20)$

Solution:
$f(x,y)=2x+y-3$
So, $f(5,20)=2(5)+20-3=27$

3) $f(-1,2)$

Solution:
$f(x,y)=2x+y-3$
So, $f(-1,2)=2(-2)+2-3=-5$

4) $f(4,2)$

Solution:
$f(x,y)=2x+y-3$
So, $f(4,2)=2(3)+2-3=5$

The Domain and Range of a Function of Two Variables

Example: Find the domain of $f(x,y)={\sqrt {9-x^{2}-y^{2}}}$

Notice that : The radicand should be non-negative. So, $9-x^{2}-y^{2}\geq 0$, hence the domain is $x^{2}+y^{2}\leq 9$ (or the set of all point that lie inside the circle).

Notice: $x^{2}+y^{2}=9$ is the circle center at $(0,0)$, radius 3.

Since the point $(0,0)$ satisfies the inequality $x^{2}+y^{2}\leq 9$. Hence the range is $0\leq x\leq 3$

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