Functions of Several Variables (Introduction)

Houston Math Prep introduces multi-variable functions, explaining the concept of a function of more than one variable and how to visualize their domains. The video clarifies the difference between single-variable functions, which produce a single y-value for each x-value, and multi-variable functions, which require ordered pairs (x, y) to produce a z-value. The domain of a multi-variable function is a set of points in the xy-plane, and the video uses the vertical line test in 3D space to illustrate the concept of a function of two variables. An example of a sphere is given to show a shape that is not a function of x and y due to the possibility of multiple z-values for the same x and y.

The script discusses how to determine the domain of functions with two variables, using the example of the upper hemisphere of a sphere to illustrate the concept. The domain is visualized as the shadow cast by the hemisphere on the xy-plane, which in this case is a solid disk of radius 3. The video explains that the domain is the set of all points in the xy-plane that, when plugged into the function, yield a defined real number for z. Points outside this region would result in an undefined value due to the square root of a negative number, which is not a real number.

The video provides a step-by-step guide on how to find and sketch the domains of various functions of two variables. It covers functions involving square roots, where the domain is determined by ensuring the expressions under the square roots are non-negative. The script uses examples such as the square root of x plus the square root of y, and the square root of 4 minus x squared minus y, to demonstrate how to set up domain statements and sketch the regions in the xy-plane where the function is defined.

The script addresses the restrictions imposed by division and square roots in functions of two variables. For fractions, the denominator must not be zero, which is translated into the condition that x plus y should not equal zero for the domain. For square roots, the expression under the root must be non-negative. The video uses examples like the function 1 over the square root of 1 minus x squared minus y squared to show how to avoid these restrictions, resulting in a domain that excludes a line through the origin with a slope of negative one.

The video concludes with examples of functions involving exponentials and logarithms, which have fewer restrictions on their domains. For an exponential function like e to the negative x squared minus y squared, any real number values for x and y are acceptable, resulting in the entire xy-plane as the domain. For a logarithmic function, the argument must be positive, leading to a domain where one variable is greater than the other. The script emphasizes that understanding the operations within the function is key to determining the domain.

In the final paragraph, Houston Math Prep wraps up the discussion on functions of more than one variable and finding their domains. The video has covered various examples and methods for determining where these functions are defined in the xy-plane. The host thanks viewers for watching and teases the next video in the series.