Find f + g, f − g, fg, and f/g and their domains: f(x) = x - 6 and g(x)  = 5x².

Functions are very important concepts in mathematics that form the backbone of topics like calculus. The fundamental operations can be performed on two or more functions to give a new function as a result.

Answer: For f(x) = x - 6 and g(x) = 5x², we get: f(x) + g(x) = x - 6 + 5x², f(x) - g(x) = x - 6 - 5x², f(x).g(x) = 5x³ - 30x², f(x) / g(x) = (x - 6) / 5x². 

Let's understand the solution in detail.

Explanation:

We can perform the fundamental operations on functions f and g.

Addition: f(x) + g(x) = x - 6 + 5x²

Since the resultant function has a finite value for all values of x, the domain of f + g is all the real numbers.

Subtraction: f(x) - g(x) = x - 6 - 5x²

Since the resultant function has a finite value for all values of x, the domain of f - g is all the real numbers.

Multiplication: f(x).g(x) = (x - 6) × 5x² = 5x³ - 30x².

Since the resultant function has a finite value for all values of x, the domain of fg is all the real numbers.

Division: f(x) / g(x) = (x - 6) / 5x²

We can see that, if x = 0, then the resultant function becomes undefined. Hence, the domain of f/g is all the real numbers except zero.

Hence, for f(x) = x - 6 and g(x) = 5x², we get: f(x) + g(x) = x - 6 + 5x², f(x) - g(x) = x - 6 - 5x², f(x).g(x) = 5x³ - 30x², f(x) / g(x) = (x - 6) / 5x².