Homogeneous Functions Part 1

Summary

The lesson introduces homogeneous functions. A function is homogeneous of degree (k) if multiplying every variable by a factor (a) multiplies the function’s value by (a^k):

[ f(ax_1, ax_2, \ldots, ax_n) = a^k f(x_1, x_2, \ldots, x_n). ]

Here, (a) is the scaling factor, and (k) is the function’s degree.

Example

For (f(x) = 7x^2), replace (x) with (3x):

[ f(3x) = 7(3x)^2 = 9(7x^2) = 3^2 f(x). ]

Since scaling (x) by (3) scales the function by (3^2), (f(x) = 7x^2) is homogeneous of degree 2.

Method for Identifying the Degree

1. Scale each variable by the same factor (a).
2. Substitute the scaled variables into the function.
3. Compare the result with the original function.
4. If the result is (a^k) times the original function, the function is homogeneous of degree (k).

Speaker and Source

• Joven Ramirez — instructor and sole speaker featured.