Problem 8 - Calculus for Biology and Medicine

Question

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=1-1 / x $$

Step-by-Step Solution

Answer

The graph has a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 1 \), crossing the x-axis at \( (1,0) \).

Step 1: Determine the Domain

The function given is \( y = 1 - \frac{1}{x} \). The largest possible domain for this function is all real numbers except where the denominator is zero. The denominator is zero when \( x = 0 \). Hence, the domain is all real numbers except \( x = 0 \).

Step 2: Find the Asymptotes

Since the function has a form of \( y = a - \frac{1}{x} \), there will be vertical and horizontal asymptotes. The vertical asymptote is at \( x = 0 \), where the function is undefined. For the horizontal asymptote, as \( x \to \infty \) or \( x \to -\infty \), \( \frac{1}{x} \to 0 \), thus \( y \to 1 \). So, there is a horizontal asymptote at \( y = 1 \).

Step 3: Determine Intercepts

To find the y-intercept, set \( x = 1 \) in the function: \( y = 1 - \frac{1}{1} = 0 \). So, the intersection with the y-axis is at (1, 0). To find the x-intercept, set \( y = 0 \): \( 1 - \frac{1}{x} = 0 \rightarrow \frac{1}{x} = 1 \rightarrow x = 1 \), thus the intersection with the x-axis is at (1, 0).

Step 4: Analyze the Behavior of the Function

Analyze how \( y \) behaves as \( x \) approaches zero from both negative and positive sides: As \( x \to 0^+ \), \( y = 1 - \frac{1}{x} \to -\infty \). As \( x \to 0^- \), \( y = 1 - \frac{1}{x} \to \infty \). Also, check behavior at infinities: As \( x \to \infty \) or \( x \to -\infty \), \( y \to 1 \).

Step 5: Draw the Graph

Using all the information: domain, intercepts, asymptotes, and function behavior: start plotting. From left of the y-axis (approaching 0 from the negative direction), the curve approaches \( y = 1 \) from above and shoots up; from the right of the y-axis (approaching 0 from the positive direction), it starts at \( y = 1 \) from below and dives down. The curve crosses the x-axis at (1,0), follows the asymptote \( y = 1 \) as \( x \to \infty \) in both directions.

Key Concepts

Asymptotes

Asymptotes are lines that a graph approaches but never truly touches or crosses. In the realm of rational functions, asymptotes serve as guides, showing the behavior of the graph as the variable tends towards very large positive or negative values or certain points of undefined values.

For the function, \( y = 1 - \frac{1}{x} \), we identify both vertical and horizontal asymptotes:

• Vertical Asymptote: This occurs at \( x = 0 \), where the function becomes undefined due to division by zero. As \( x \) approaches zero from either side, the function tends towards infinity or negative infinity, never actually reaching a value.
• Horizontal Asymptote: As \( x \) moves towards very large positive or negative values (\( x \to \infty \) or \( x \to -\infty \)), the \( \frac{1}{x} \) term diminishes to zero, pulling the function's value creepily closer to \( y = 1 \), hence the horizontal asymptote is \( y = 1 \).

These asymptotes are crucial as they fundamentally guide the shape and behavior of the graph across its domain.

Intercepts

Intercepts are simply the points where a graph crosses the axes. Understanding how to find these points offers insight into the interaction between the function and the coordinate plane.

For this function, let's identify both types of intercepts:

• Y-Intercept: This is found by setting \( x = 0 \) in the equation of the function, which gives us \( y = 1 - \frac{1}{x} \). However, as we noted, the function is undefined at \( x = 0 \), thus technically, there is no y-intercept! This is an important distinction as it highlights the effect of the vertical asymptote.
• X-Intercept: Setting \( y = 0 \) allows us to solve \( 1 - \frac{1}{x} = 0 \). Solving yields \( \frac{1}{x} = 1 \), so \( x = 1 \). Thus, the graph crosses the x-axis at the point (1,0).

Locating intercepts is key to sketching an accurate graph of the rational function by providing anchor points.

Domain of a Function

Understanding the domain of a function is critical to graphing and analyzing any mathematical function. The domain represents all possible input values (or \( x \) values) for which the function is defined.

In the case of the function \( y = 1 - \frac{1}{x} \), the domain includes all real numbers except where the function is not defined due to division by zero. Here, this is evident where \( x = 0 \) makes the function undefined.

• Thus, the domain of \( y = 1 - \frac{1}{x} \) can be expressed as \( x \in \mathbb{R}, \, x eq 0 \).

By fully understanding the domain, we've established the range of input values possible, which aids in predicting how the graph behaves and can be sketched and analyzed accurately.