$\frac{11-x}{(2 x-1)(x+3)}$
Answer (1)
Domain: All real numbers except x = 2 1 and x = − 3 .
x-intercept: x = 11 .
y-intercept: y = − 3 11 .
Vertical asymptotes: x = 2 1 and x = − 3 .
Horizontal asymptote: y = 0 .
Explanation
Analyzing the Expression We are given the rational expression ( 2 x − 1 ) ( x + 3 ) 11 − x . We will analyze this expression by finding its domain, intercepts, and asymptotes.
Finding the Domain First, let's find the domain. The domain consists of all real numbers except for those that make the denominator equal to zero. So, we need to solve ( 2 x − 1 ) ( x + 3 ) = 0 . This gives us 2 x − 1 = 0 or x + 3 = 0 . Solving these equations, we get x = 2 1 or x = − 3 . Therefore, the domain is all real numbers except x = 2 1 and x = − 3 .
Finding the x-intercept Next, let's find the x-intercept. The x-intercept occurs when the numerator is equal to zero. So, we need to solve 11 − x = 0 , which gives us x = 11 . Thus, the x-intercept is x = 11 .
Finding the y-intercept Now, let's find the y-intercept. The y-intercept occurs when x = 0 . Substituting x = 0 into the expression, we get ( 2 ( 0 ) − 1 ) ( 0 + 3 ) 11 − 0 = ( − 1 ) ( 3 ) 11 = − 3 11 . Thus, the y-intercept is y = − 3 11 .
Finding the Vertical Asymptotes Let's determine the vertical asymptotes. Vertical asymptotes occur at the values of x that make the denominator zero, provided the numerator is not also zero at those values. We already found that the denominator is zero at x = 2 1 and x = − 3 . Since the numerator is not zero at these values, the vertical asymptotes are x = 2 1 and x = − 3 .
Finding the Horizontal Asymptote Now, let's determine the horizontal asymptote. Since the degree of the numerator (1) is less than the degree of the denominator (2), the horizontal asymptote is y = 0 .
Summary of Findings In summary:
Domain: All real numbers except x = 2 1 and x = − 3 .
x-intercept: x = 11
y-intercept: y = − 3 11
Vertical asymptotes: x = 2 1 and x = − 3
Horizontal asymptote: y = 0
Examples
Rational expressions are used in various fields, such as physics and engineering, to model real-world phenomena. For example, in electrical engineering, rational functions can describe the impedance of a circuit as a function of frequency. Analyzing the asymptotes and intercepts of these functions helps engineers understand the behavior of the circuit at different frequencies and design circuits that meet specific performance requirements. Understanding the behavior of rational functions is crucial for analyzing and designing such systems.
