# Match the function with its graph (labeled i-vi)

– $f(x,y) = |x| + |y|$

– $f(x,y) = |xy|$

– $f(x,y) = \frac{1}{1+x^2+y^2}$

– $f(x,y) = (x^2 – y^2)^2$

– $f(x,y) =(x-y)^2$

– $f(x,y) = sin (|x| + |y|)$

This question aims to find the best graph match for the given functions by using the concepts of Calculus.

This question uses the basic concepts of Calculus and linear algebra by matching the functions to the best contour graphs. Contour graphs simply map the two-dimension input function and output function of one dimension. The basic figure of the contour graph is shown below:

a)$f(x,y) = |x| + |y|$:

Suppose f(x,y)  is equal to  Z, then we have Z equal to |x| when the value of y is zero while Z is equal to |y| when the value of x is zero. So for this equation, the best graph is labeled VI.

b) $f(x,y) = |xy|$:

Suppose f(x,y)  is equal to  Z, then we have Z equal to zero when the value of y is zero while Z is equal to zero when the value of x is zero. So for this equation, the best graph is labeled V.

c) $f(x,y) = \frac{1}{1+x^2+y^2}$:

Suppose f(x,y) is equal to  Z, so when the value of x is zero, we get

$\frac{1}{1+y^2}$

and when the value of y is zero, then we have:

$\frac{1}{1+x^2}$

When the value of x  and y is very large, it will result in a zero value for Z so the best match graph is I.

d) $f(x,y) = (x^2 – y^2)^2$:

Suppose f(x,y)  is equal to  Z, then the value of x is zero, we have:

$Z=y^4$

and when the value of y is zero, we have:

$Z=x^4$

and if Z is equal to zero then:

$y=x$

so the best graph match is IV.

e) $f(x,y) =(x-y)^2$:

Suppose f(x,y)  is equal to  Z, then the value of x is zero, we have:

$Z=y^2$

and when the value of y is zero, we have:

$Z=x^2$

and if Z is equal to zero then:

$y=x$

so the best graph match is II.

f) $f(x,y) = sin (|x| + |y|)$:

Suppose f(x,y)  is equal to  Z, then the value of x is zero, we have:

$sin(|y|)$

and when the value of y is zero, we have:

$sin(|x|)$

so the best graph match is III.

## Numeric Result

By assuming the values of $x$ and $y$, the given functions are matched in the best contour graph.

## Example

Draw the graph for function $f(x,y) = cos(|x|+|y|)$.

Suppose f(x,y)  is equal to  Z, then the value of x is zero, we have:

$cos(|y|)$

and when the value of y is zero, we have:

$cos(|x|)$

so the best graph  for the given function is as follows:

Images/Mathematical drawings are created with Geogebra.