SCALAR-VALUED MULTIVARIABLE FUNCTIONS

A function $f:D \rightarrow ℝ$ is a scalar-valued multivariable function iff $D \subseteq ℝ^n$, where $n>1$.

• At least 2 parameters as input
• The output is a real number

SOME EXAMPLES

• $f(x,y)=x^2+y^2$
• $f(x,y,z)=ln(x+y)-3z^2+yz$
• $f(x,y)=\sin(xy)$
• $f(x,y,z,w)=3x+4y+5z+6w^3$

PLOTTING GRAPHS

If the domain is a subset of $ℝ^2$, then

• $XY$ - $Plane$ : Domain
• $z$ - $axis$ : Codomain

Let us look at the graphs of a few functions.

$f(x,y)=x+y$

$(5,5)$ is in the domain. $5+5=10$ is in the codomain. The graph contains the point $(5,5,10)$. This graph is also a plane in $ℝ^3$

$f(x,y)=\sin(xy)$

This graph is oscillating. Notice the function along the $x$ - $axis$ and the $y$ - $axis$. The value of $xy$ is $0$ along these two axes, as a result, $\sin(xy)$ is also $0$. Therefore, the graph is touching the $XY$ - $Plane$ along these two axes.

$f(x,y)=x^2+y^2$

$x^2+y^2$ is a sum of two squared terms. It is always greater than or equal to $0$. It will be equal to $0$ only when $x=0$ and $y=0$