Understanding the Domain and Range of Functions: The Case of (f(x) frac{x - 2}{2 - x})

Understanding the domain and range of functions is essential for analyzing and interpreting mathematical functions. This article will delve into the specific case of the function (f(x) frac{x - 2}{2 - x}) and explore its domain and range in detail. We will provide a thorough explanation and use relevant mathematical principles to ensure clarity and depth.

Introduction

The function (f(x) frac{x - 2}{2 - x}) is an interesting case to study due to the presence of a denominator that restricts the values of (x). By examining this function, we can see how the domain and range interact to define the function's behavior.

Defining the Domain and Range

The domain of a function is the set of all possible input values (i.e., the values of (x)) for which the function is defined. The range is the set of all possible output values (i.e., the values of (f(x))) that the function can produce. For the function (f(x) frac{x - 2}{2 - x}), we need to identify the values of (x) for which the expression is defined and the corresponding values of (f(x)).

Determining the Domain

The domain is determined by ensuring that the denominator is not zero, as division by zero is undefined. Therefore, we need to solve the equation (2 - x eq 0).

2 - x ≠ 0

(x eq 2)

Hence, the domain of (f(x)) is all real numbers except (x 2). In set notation, this is written as:

D_f mathbb{R} - {2}

This means that the function (f(x) frac{x - 2}{2 - x}) is defined for all real numbers (x) except (x 2).

Determining the Range

To find the range, we need to analyze the possible values of (f(x)) for the input values in the domain. We start by simplifying the function:

f(x) frac{x - 2}{2 - x} frac{x - 2}{-(x - 2)} -1

The expression simplifies to (-1) for all (x eq 2). Therefore, the function (f(x)) is a constant function with a value of (-1) for all inputs in the domain (x eq 2).

From this, we can clearly see that the range of (f(x)) is simply the set containing the single value (-1). In set notation, this is written as:

R_f {-1}

This means that the function (f(x) frac{x - 2}{2 - x}) always outputs (-1) for any input in the domain.

Additional Insights

Let's further explore the behavior of the function. We can consider specific values of (x) to see how the function behaves:

When (x 0):f(x) frac{0 - 2}{2 - 0} frac{-2}{2} -1 When (x 1):f(x) frac{1 - 2}{2 - 1} frac{-1}{1} -1 When (x 3):f(x) frac{3 - 2}{2 - 3} frac{1}{-1} -1

As we can see, regardless of the value of (x) (as long as (x eq 2)), the output is always (-1). This confirms that the range is indeed ({-1}).

Conclusion

In summary, for the function (f(x) frac{x - 2}{2 - x}), the domain is all real numbers except (x 2) and the range is ({-1}). This function is a constant function with a value of (-1) for all inputs except (x 2).

Understanding the domain and range of functions is crucial for a comprehensive analysis. By examining the specific case of (f(x) frac{x - 2}{2 - x}), we have demonstrated how to determine the domain and range, and how the function behaves as a constant function.

Keywords

domain: The set of all possible input values for a function.

range: The set of all possible output values of a function.

function analysis: The study of mathematical functions to understand their behavior.

real numbers: All rational and irrational numbers that can be represented on a number line.