We're learning about different kinds of functions and I don't really understand the difference between rational and algebraic functions. I know that rational functions are functions that are a ratio of two polynomials, and algebraic functions are any functions that can be made with multiplication, division, addition, subtraction, taking the root of, etc. polynomials. All rational functions are algebraic, but not the other way around, correct? So does that mean that the function x-6/x+6 is both algebraic and rational?

Yes, and yes.

Yes, you're correct! Rational functions are indeed functions that can be expressed as the ratio of two polynomials. Algebraic functions, on the other hand, include not only rational functions but also any functions that can be constructed using polynomials and the basic operations of arithmetic, such as addition, subtraction, multiplication, division, and taking roots.

In the case of the function f(x) = (x - 6)/(x + 6), it is both algebraic and rational. It is algebraic because it can be constructed using the operations of arithmetic: subtraction, addition, and division. It is rational because it is indeed a ratio of two polynomials, namely (x - 6) and (x + 6).

Remember that rational functions have the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomials, and q(x) ≠ 0. Since the function you provided follows this format, it is a rational function.

To determine if a function is algebraic or rational, you can analyze its structure and see if it can be expressed as a ratio of polynomials (in the case of rational functions) or if it can be constructed using polynomials and basic arithmetic operations (in the case of algebraic functions).