Problem 11 - A Graphical Approach to Precalculus with Limits

Question

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. State the domain of \(f .\) $$f(x)=\frac{4-3 x}{2 x+1}$$

Step-by-Step Solution

Answer

Vertical asymptote: \(x = -\frac{1}{2}\); horizontal asymptote: \(y = -\frac{3}{2}\); domain: \(x \in \mathbb{R}, x \neq -\frac{1}{2}\).

Step 1: Identify Asymptotes

For a rational function \(f(x) = \frac{a(x)}{b(x)}\), vertical asymptotes occur where the denominator \(b(x)\) equals zero (provided that the numerator \(a(x)\) does not also equal zero at those points). Set the denominator equal to zero: \[2x + 1 = 0\] Solve for \(x\): \[x = -\frac{1}{2}\]This provides us with one vertical asymptote at \(x = -\frac{1}{2}\).

Step 2: Identify Horizontal or Oblique Asymptotes

Horizontal or oblique asymptotes of a rational function are determined by the degrees of the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at \(y = 0\). In this case, the numerator \(4 - 3x\) is degree 1 and the denominator \(2x + 1\) is degree 1 (same degrees). When the degrees are the same, divide the leading coefficients: \[\frac{-3}{2} = -\frac{3}{2}\]Thus, the horizontal asymptote is \(y = -\frac{3}{2}\).

Step 3: State Domain and Confirm No Oblique Asymptotes

The domain of \(f(x)\) is all real numbers except where the denominator is zero. Since the vertical asymptote is at \(x = -\frac{1}{2}\), the domain is \(x \in \mathbb{R}, x eq -\frac{1}{2}\). There are no oblique asymptotes since the degrees of the numerator and denominator are equal.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur in the graph of a rational function whenever the denominator equals zero and the numerator does not equal zero at those points.
To find these vertical asymptotes, set the denominator of the rational function equal to zero and solve for the variable.
In our example, given the function \( f(x) = \frac{4 - 3x}{2x + 1} \), we identify the vertical asymptote by solving the equation \( 2x + 1 = 0 \).
This equation simplifies to \( x = -\frac{1}{2} \).
Thus, there is a vertical asymptote at \( x = -\frac{1}{2} \).
Remember that vertical asymptotes describe the values of \( x \) that are not in the domain of the function, as the function approaches positive or negative infinity near these vertical lines.

Horizontal Asymptotes

Horizontal asymptotes in rational functions depend on the degrees of the polynomials in the numerator and the denominator.
The degree of a polynomial is the highest power of \(x\) in the polynomial.
Here are the rules for determining horizontal asymptotes:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
• If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients of the numerator and denominator, respectively.
• There is no horizontal asymptote if the degree of the numerator is greater.

For \( f(x) = \frac{4 - 3x}{2x + 1} \), both the numerator and denominator are of degree 1.
Therefore, the horizontal asymptote is determined by \( \frac{-3}{2} \), which makes it \( y = -\frac{3}{2} \).
This horizontal line is what the graph approaches as \( x \) moves towards infinity or negative infinity.

Rational Functions

A rational function is a function that is expressed as the ratio of two polynomials.
An example of a rational function is \( f(x) = \frac{4 - 3x}{2x + 1} \), where \( 4 - 3x \) is the numerator and \( 2x + 1 \) is the denominator.
Key characteristics of rational functions include their potential asymptotes and domains.

**Finding the Domain**

The domain of a rational function includes all real numbers except those that make the denominator zero.
To find these values, set the denominator equal to zero and solve for \( x \).
From the function \( f(x) \), the denominator \( 2x + 1 = 0 \) gives \( x = -\frac{1}{2} \).
The domain of this function is all real numbers except \( x = -\frac{1}{2} \).

**Key Points**

• The degrees of polynomials in the rational function determine if there will be horizontal or oblique asymptotes, and vertical asymptotes provide the points where the function is undefined.
• Always identify and state these aspects to fully understand the behavior of the rational function and its graph.