# Find f + g, f − g, fg, and f/g and their domains: f(x) = x - 6 and g(x) = 5x^{2}.

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) = 5x^{2}, we get: f(x) + g(x) = x - 6 + 5x^{2}, f(x) - g(x) = x - 6 - 5x^{2}, f(x).g(x) = 5x^{3} - 30x^{2}, f(x) / g(x) = (x - 6) / 5x^{2}.

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 + 5x^{2}

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 - 5x^{2}

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) × 5x^{2} = 5x^{3} - 30x^{2}.

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) / 5x^{2}

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.