Problem 44 - Precalculus Mathematics for Calculus

Question

Find the intercepts and asymptotes, and then sketch a graph of the rational function and state the domain and range. Use a graphing device to confirm your answer. $$s(x)=\frac{1-2 x}{2 x+3}$$

Step-by-Step Solution

Answer

Intercepts: (0,1/3) and (1/2,0). Vertical asymptote: x = -3/2. Horizontal asymptote: y = -1. Domain: x ≠ -3/2. Range: y ≠ -1.

Step 1: Find the y-intercept

To find the y-intercept, set the value of \(x\) to 0 and solve for \(s(x)\). Substituting in the equation: \(s(0) = \frac{1-2 \times 0}{2 \times 0 + 3} = \frac{1}{3}\). Therefore, the y-intercept is \((0, \frac{1}{3})\).

Step 2: Find the x-intercept

To find the x-intercept, set the numerator of \(s(x)\) to zero and solve for \(x\). The equation is: \(1 - 2x = 0\). Solve this to get \(x = \frac{1}{2}\). Therefore, the x-intercept is \((\frac{1}{2}, 0)\).

Step 3: Find the vertical asymptote

To find the vertical asymptote, set the denominator equal to zero and solve for \(x\). The equation is: \(2x + 3 = 0\). Solve this to get \(x = -\frac{3}{2}\). Hence, there is a vertical asymptote at \(x = -\frac{3}{2}\).

Step 4: Find the horizontal asymptote

For large values of \(|x|\), the horizontal asymptote is determined by comparing the degrees of the numerator and denominator. Since both have degree 1, the horizontal asymptote is \(y = \frac{-2}{2} = -1\).

Step 5: Sketch the Graph

Using the intercepts and asymptotes, sketch the rational function. \(s(x)\) approaches the horizontal asymptote \(y = -1\) as \(x\) approaches infinity, and the curve crosses at the x-intercept \((\frac{1}{2}, 0)\) and y-intercept \((0, \frac{1}{3})\). Remember, the curve does not intersect the vertical asymptote at \(x = -\frac{3}{2}\).

Step 6: State the Domain and Range

The domain is all real numbers except \(x = -\frac{3}{2}\) (where the vertical asymptote is). Therefore, the domain is \(x \in \mathbb{R} \setminus \{-\frac{3}{2}\}\). The range is all real numbers except \(y = -1\) (where the horizontal asymptote is). Thus, the range is \(y \in \mathbb{R} \setminus \{-1\}\).

Key Concepts

Intercepts

Intercepts are the key points where a graph intersects the axes. Let's break down how to find both x and y intercepts of a rational function like \(s(x)=\frac{1-2x}{2x+3}\).
- **Y-Intercept**: To find the y-intercept, we set \(x\) to zero. This simplifies our equation: \(s(0) = \frac{1}{3}\). Thus, the y-intercept is at the point \((0, \frac{1}{3})\). This means the graph crosses the y-axis at this point.
- **X-Intercept**: Finding the x-intercept involves setting the numerator equal to zero. This gives us the equation \(1 - 2x = 0\). Solving this, we discover \(x = \frac{1}{2}\), meaning the graph crosses the x-axis at \((\frac{1}{2}, 0)\).
These intercepts provide valuable checkpoints for graphing the rational function, illustrating where the graph actually meets the coordinate axes.

Asymptotes

Asymptotes are straight lines that the graph of a function approaches but never actually touches. Identifying these helps us anticipate the behavior of the graph at extreme values. Rational functions can have both vertical and horizontal asymptotes.
- **Vertical Asymptote**: This occurs where the denominator equals zero, since division by zero is undefined. For \(s(x) = \frac{1-2x}{2x+3}\), setting the denominator to zero gives \(2x + 3 = 0\), leading to \(x = -\frac{3}{2}\). The graph will approach but never touch this vertical line.
- **Horizontal Asymptote**: This shows the behavior of \(s(x)\) as \(x\) approaches infinity or negative infinity. Since the degrees of the numerator and denominator are equal (both 1), the horizontal asymptote is determined by the leading coefficients. Here it's \(y = -1\). This tells us that as \(x\) moves towards infinity or negative infinity, \(s(x)\) hovers near this horizontal line.
Knowing asymptotes allows us to sketch more accurate graphs and better understand the function's long-term tendencies.

Domain and Range

Understanding the domain and range of a rational function is crucial. They tell us what inputs (x-values) the function can accept and what outputs (y-values) it can produce.
- **Domain**: This refers to all possible x-values for which the function is defined. Since division by zero is not defined, our domain for \(s(x) = \frac{1-2x}{2x+3}\) excludes the x-value where the denominator is zero, so \(x eq -\frac{3}{2}\). Therefore, the domain in set notation is \(x \in \mathbb{R} \setminus \{-\frac{3}{2}\}\).
- **Range**: This encompasses all possible y-values the function can output. Referring to our horizontal asymptote \(y = -1\), the function will not reach this value as well. Thus, the range of \(s(x)\) is \(y \in \mathbb{R} \setminus \{-1\}\).
By grasping domain and range, one can better predict and visualize function behaviors, ensuring a comprehensive understanding of the rational function's scope and limitations.