1 Functions of several variables

We know that $f\left(x\right)$ is used to represent a function of one variable: the input variable is $x$ and the output is the value $f\left(x\right)$ . Here $x$ is the independent variable and $y=f\left(x\right)$ is the dependent variable .

Suppose we consider a function with two independent input variables $x$ and $y$ , for example

$f\left(x,y\right)=x+2y+3.$

If we specify values for $x$ and $y$ then we have a single value $f\left(x,y\right)$ . For example, if $x=3$ and $y=1$ then $f\left(x,y\right)=3+2+3=8$ . We write $f\left(3,1\right)=8$ .

Task!

Find the values of $f\left(2,1\right),f\left(-1,-3\right)$ and $f\left(0,0\right)$ for the following functions.

1. $f\left(x,y\right)=x^2+y^2+1$
2. $f\left(x,y\right)=2x+xy+y^3$

Answer

In a similar way we can define a function of three independent variables. Let these variables be $x,y$ and $u$ and the function $f\left(x,y,u\right)$ .

Example 1

Given $f\left(x,y,u\right)=x^2+yu+2$ , find $f\left(0,1,0\right),f\left(-1,-1,2\right)$ .

Solution

$f\left(0,1,0\right)=0^2+1\times 0+2=2;f\left(-1,-1,2\right)=1-2+2=1$

Task!

1. Find $f\left(2,-1,1\right)$ for $f\left(x,y,u\right)=xy+yu+ux$ .
2. Evaluate $f\left(x,y,u,t\right)=x^2-y^2-u^2-2t$ when $x=1,y=-2,u=3,t=1.$

Answer