Finding the domain of a function is an important step in understanding its behavior and limitations. In order to find the domain of a function, we need to determine all values of the independent variable for which the function is defined. Two common ways to find the domain of a function are graphing and algebraic manipulation.

Let’s consider the function a(x) = 3x + 1. To find its domain, we need to determine all values of x for which a(x) is defined. Since a(x) is a linear function, it is defined for all real numbers. Therefore, the domain of a(x) is the set of all real numbers, which we can write as:

Domain(a) = (-∞, ∞)

Another way to check this result is by using algebraic manipulation. Recall that the expression 3x + 1 is defined for all real numbers. The only restriction on x is that we cannot divide by zero or take the square root of a negative number. Since neither of these operations appear in the expression for a(x), we conclude that the domain of a(x) is all real numbers.

In conclusion, finding the domain of a function is a crucial step in analyzing its properties. We saw how to find the domain of the linear function a(x) = 3x + 1 both by graphing and by algebraic manipulation. By understanding the domain of a function, we can avoid making mistakes and ensure that our analysis is accurate.