Understanding the Domain and Range of y (2x - 1)/(x - 7)

When dealing with rational functions, identifying the domain and range is a fundamental step. This article will guide you through the process using the function y (2x - 1)/(x - 7).

What is a Rational Function?

A rational function is a function that can be expressed as the quotient or ratio of two polynomials. The general form is f(x) P(x)/Q(x), where P(x) and Q(x) are polynomials.

Domain of y (2x - 1)/(x - 7)

The domain of a rational function is all real numbers except where the denominator is zero. To find this, we set the denominator equal to zero and solve for x:

$$x - 7 0$$ $$x 7$$

Therefore, the domain of the function y (2x - 1)/(x - 7) is all real numbers except x 7. We can represent this as:

$$ (-infty, 7) cup (7, infty) $$

Range of y (2x - 1)/(x - 7)

To find the range, we need to determine the values that y can take. We start by finding the inverse function and then determining its domain, which will be the range of the original function.

Step 1: Find the Inverse Function

We will interchange x and y and solve for y:

$$ x frac{2y - 1}{y - 7} $$ $$ y - 7 frac{2x - 1}{x} $$ $$ y - 7x 2x - 1 $$ $$ y frac{7x - 1}{x - 2} $$

The resulting function is y (7x - 1)/(x - 2).

Step 2: Determine the Domain of the Inverse Function

The domain of the inverse function will be the range of the original function. Setting the denominator equal to zero and solving for x gives:

$$ x - 2 0 $$ $$ x 2 $$

Therefore, the range of the function y (2x - 1)/(x - 7) is all real numbers except y 2. We can represent this as:

$$ (-infty, 2) cup (2, infty) $$

Graphical Interpretation

The function y (2x - 1)/(x - 7) has a vertical asymptote at x 7 and a horizontal asymptote at y 2.

Vertical Asymptote: When x approaches 7, the function y approaches either positive or negative infinity, indicating a vertical asymptote at x 7.

Horizontal Asymptote: As x approaches infinity or negative infinity, y approaches 2, indicating a horizontal asymptote at y 2.

Conclusion

In summary, the domain of the rational function y (2x - 1)/(x - 7) is all real numbers except x 7, and the range is all real numbers except y 2. Understanding these concepts is crucial for analyzing and graphing rational functions, which are fundamental in many areas of mathematics and its applications.