Understanding the Domain and Range of the Rational Function f(x) 3/(x-1)

When working with rational functions, it is crucial to understand the domain and range. This article will guide you through determining the domain and range of the given rational function f(x) 3/(x-1). Understanding these concepts will not only enhance your skills in algebra but also help you visualize the function's behavior on a graph.

Domain

The domain of a function encompasses all the input values (x) for which the function is defined. For rational functions, the function becomes undefined where the denominator is zero. Let's determine the domain of f(x) 3/(x-1) by setting the denominator to zero and solving for x.

Let's solve for x:

x - 1 0 implies x 1

This means that when x 1, the denominator is zero, making the function undefined. Therefore, the domain of f(x) 3/(x-1) includes all real numbers except x 1.

The domain can be expressed as:

Domain: (-∞, 1) ∪ (1, ∞)

Range

The range of a function encompasses all the output values (y) that the function can take. To determine the range of f(x) 3/(x-1), we need to consider the values of y for which the function is defined. It's important to note that a rational function can never produce exactly the value that would make the numerator zero.

To find when fx 0:

3/(x-1) 0

This equation has no solution because a fraction is zero only when both the numerator and the denominator are zero. However, here the numerator is a constant 3, which cannot be zero. Therefore, fx 0 is never possible. As a result, the function f(x) 3/(x-1) can take any real value except zero.

The range can be expressed as:

Range: (-∞, 0) ∪ (0, ∞)

Graphical Representation

The graphical representation of the function f(x) 3/(x-1) reveals its domain and range. As mentioned, the function is undefined at x 1, which is marked by a vertical asymptote. The function's behavior around this asymptote can be seen in the graph, where the function approaches negative infinity as x approaches 1 from the left and positive infinity as x approaches 1 from the right.

The horizontal asymptote of this function can be found by considering the behavior of the function as x approaches positive and negative infinity. As x becomes very large, the term 1 in the denominator becomes negligible, and the function approaches zero from both positive and negative directions.

Conclusion

Understanding the domain and range of a rational function like f(x) 3/(x-1) is essential for comprehending its behavior and graph. The domain of this function is all real numbers except x 1, while the range includes all real numbers except zero.

Key Takeaways: The domain is (-∞, 1) ∪ (1, ∞). The range is (-∞, 0) ∪ (0, ∞). A vertical asymptote exists at x 1, and a horizontal asymptote exists at y 0.

Further Exploration

For those interested in further exploring the properties of rational functions, consider experimenting with different constants and denominators. Understanding how these elements affect the domain and range can be both educational and insightful.