Match each of graphs (A) and (B) in Figure 21 with one of...

Match each of graphs (A) and (B) in Figure 21 with one of the following functions:
$$
\begin{array}{l}{\text { (i) } f(x, y)=(\cos x)(\cos y)} \\ {\text { (ii) } g(x, y)=\cos \left(x^{2}+y^{2}\right)}\end{array}
$$

Key Concepts

Radial Symmetry

Radial symmetry refers to a situation where a function’s value depends only on the distance from a central point, typically the origin, rather than the individual x and y coordinates. When a function is expressed in terms of x² + y², it implies that points equidistant from the origin yield the same function value. This concept is key in recognizing and interpreting circular patterns or ring-like structures in the graph of a function.

Trigonometric Functions in Multiple Variables

Trigonometric functions such as cosine, when extended to two variables, can produce intricate oscillatory patterns depending on how the arguments are formulated. For example, composing cosine with independent variables or with a radial term like x² + y² leads to distinct surface features. Understanding the periodic and oscillatory nature of these functions is essential for matching analytical expressions with their corresponding graphs.

Graphing Functions of Two Variables

This concept involves representing a function of two independent variables as a surface in three dimensions. It is crucial to understand how the input values (x and y) map to an output value f(x, y) to visualize or interpret the shape and behavior of the surface. This is a core idea in multivariable calculus, especially when associating analytic expressions with their geometric representations.

Separable Functions

A separable function can be written as a product or sum of functions each depending only on one variable. For example, a product of functions such as cos(x) and cos(y) is separable. This property influences the way the graph is constructed and understood, as the overall behavior in one direction can be considered independent of the behavior in the perpendicular direction.

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