Exploring the Range and Domain of a Constant Function: f(x) -12f(x)

When we encounter the function f(x) -12f(x), it's essential to understand the algebraic manipulations involved and elucidate the properties of such a function, especially the range and domain.

Introduction to the Function

The given function is intrinsically recursive, which can be simplified through basic algebraic operations. The recursive form f(x) -12f(x) implies that f(x) is related to itself in a straightforward manner, suggesting a need for a constant function as a solution.

Algebraic Simplification

Let's begin with the given function f(x) -12f(x). To simplify, we can add 12f(x) to both sides of the equation:

Add 12f(x) to both sides: This results in the equation: 13f(x) 0 Divide both sides by 13: This simplifies to: f(x) 0

The algebraic transformation above shows that the function f(x) is indeed a constant function with the value zero for all values of x.

Understanding the Range and Domain

The range of a function is the set of all possible output values (y-values) of the function. When we have a constant function such as f(x) 0, every value of x in the domain produces the same output, which is 0. Therefore, the range of the function is:

Range: {0}

In mathematical terms, the range is a singleton set containing the constant value 0. This indicates that regardless of the input values, the function will always return the value 0.

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the constant function f(x) 0, the domain is all possible values of x. Since there are no restrictions on x, the domain is the entire set of real numbers, denoted as R.

Domain: R

Thus, for any x in the set of real numbers, f(x) 0.

Conclusion

In conclusion, the function f(x) -12f(x) simplifies to a constant function f(x) 0. This function has a range of {0} and a domain of R, meaning it is defined for all real numbers and always outputs 0.

Keywords: constant function, range, domain