Illustrate by means of an example how a real-valued function...

Illustrate by means of an example how a real-valued function of the two variables $x$ and $y$ gives different real-valued functions of one variable when we restrict $y$ to be different constants.

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Key Concepts

Real-Valued Functions of Several Variables

A real-valued function of several variables is a rule that assigns a single real number to each ordered tuple of inputs from its domain. In particular, for two variables, the function f(x, y) takes pairs (x, y) from an appropriate domain (often a subset of R²) and produces a real number. This concept is fundamental in multivariable calculus and analysis, where functions depend on more than one independent variable and their behavior can change with respect to each variable.

Function Restriction (Slicing)

Function restriction, in this context, refers to fixing one of the variables in a multivariable function to obtain a new function of the remaining variable. By setting y to a constant value, for example, f(x, y) becomes a function solely of x (often denoted as f(x, c)). This process, also known as taking a slice of the function, illustrates how the variation in one variable can be examined while keeping the other variables fixed, leading to a family of functions that depend on the chosen constant value.

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