Match the function with its graph (labeled I-VI). Give...

Match the function with its graph (labeled I-VI). Give reasons for your choices.
(a) $f(x, y)=\frac{1}{1+x^{2}+y^{2}}$
(b) $f(x, y)=\frac{1}{1+x^{2} y^{2}}$
(c) $f(x, y)=\ln \left(x^{2}+y^{2}\right)$
(d) $f(x, y)=\cos \sqrt{x^{2}+y^{2}}$
(e) $f(x, y)=|x y|$
(f) $f(x, y)=\cos (x y)$

Key Concepts

Level Sets and Contour Analysis

Level sets, or contour lines, represent the set of points (x, y) where a multivariable function f(x, y) takes on the same constant value. Analyzing these level curves is a primary tool in understanding the qualitative behavior of functions, helping to visualize surfaces by looking at slices at constant function values.

Radial Symmetry and Polar Coordinates

Many functions of two variables depend on x and y only through the combination x² + y² or its square root, which implies radial symmetry. This symmetry means that the function's value depends solely on the distance from the origin, simplifying analysis and graph interpretation by viewing the behavior in terms of polar coordinates.

Behavior of Rational Functions

Rational functions, particularly those of the form 1/(1 + expression), exhibit characteristic asymptotic behavior, often decaying as the input grows large. Understanding how replacing variables with quadratic forms affects the graph is fundamental when matching functions to their surfaces.

Periodic Functions in Two Variables

Periodic behavior, as seen with cosine functions, extends into two dimensions by introducing oscillatory patterns along particular directions or radii. Recognizing periodicity in multivariable functions involves identifying repeated fluctuations in one or more directions and understanding how these patterns manifest in level curves and surfaces.

Logarithmic Function Behavior

Logarithmic functions applied to sums of squares, such as ln(x² + y²), produce surfaces with distinctive features, including singular behavior near the origin and unbounded growth as the argument increases. This understanding helps in anticipating both the local and global behavior of the function’s graph.

Absolute Value Functions and Piecewise Definitions

Functions that involve the absolute value, like |xy|, tend to have piecewise-defined behavior that can lead to ridges or sharp transitions in the graph. Recognizing the impact of absolute values is crucial for understanding symmetry and discontinuities in derivative behavior.

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