Problem 7 - Calculus for Biology and Medicine

Question

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=3+1 / x $$

Step-by-Step Solution

Answer

The graph has a vertical asymptote at \( x=0 \) and a horizontal asymptote at \( y=3 \), with an x-intercept at \((-\frac{1}{3}, 0)\).

Step 1: Identify Domain

The function given is \( y = 3 + \frac{1}{x} \). The term \( \frac{1}{x} \) is undefined when \( x = 0 \). Therefore, the domain is all real numbers except \( x = 0 \).

Step 2: Analyze the Asymptotes

There are two asymptotes for this function: a vertical asymptote at \( x = 0 \) because the function is undefined there, and a horizontal asymptote at \( y = 3 \) because as \( x \) approaches infinity, \( \frac{1}{x} \) approaches zero, making \( y \) approach 3.

Step 3: Find Intercepts

To find the y-intercept, set \( x = 0 \). As the function is undefined at \( x = 0 \), there is no y-intercept for this function. For the x-intercept, set \( y = 0 \), giving \( 3 + \frac{1}{x} = 0 \) or \( \frac{1}{x} = -3 \), leading to \( x = -\frac{1}{3} \). Thus, the x-intercept is \((-\frac{1}{3}, 0)\).

Step 4: Sketch Basic Points

Choose a few values of \( x \) to determine points on the graph. For example, when \( x = 1 \), \( y = 3 + 1 = 4 \). When \( x = -1 \), \( y = 3 - 1 = 2 \). These points are \( (1, 4) \) and \( (-1, 2) \).

Step 5: Draw the Graph

Sketch the graph approaching the vertical asymptote (x = 0) and the horizontal asymptote (y = 3). The curve will be in the first and third quadrants due to the sign of \( \frac{1}{x} \). The curve will approach but never touch the asymptotes.

Key Concepts

Domain of a Function

Understanding the **domain of a function** is crucial because it tells us where the function can "live." For the function given, which is \( y = 3 + \frac{1}{x} \), we need to identify where this function is defined.
The tricky part here is the term \( \frac{1}{x} \), which is not defined at \( x = 0 \) because division by zero is undefined in mathematics. This leads us to conclude that the domain is all real numbers except \( x = 0 \).
In practical terms:

• Any real number can be an input for \( x \),
• except for 0.
  This means the domain is: \( (-\infty, 0) \cup (0, \infty) \).
  This concept helps us outline where the graph will be drawn and where to watch for any sudden changes or breaks.

Asymptotes

Asymptotes are important features of a graph that show us where the function behaves in a certain way without ever actually reaching a specific line.
For our function \( y = 3 + \frac{1}{x} \), we have two kinds of asymptotes to consider:

• Vertical Asymptote: This occurs at \( x = 0 \). Here, the function is undefined, and the graph will approach this line very closely, without crossing it.
• Horizontal Asymptote: Here, as \( x \) extends towards infinity on either side, \( \frac{1}{x} \) starts to vanish (getting close to zero). Hence, \( y \) approaches the line \( y = 3 \) but will never equal 3.
  Asymptotes are like invisible barriers that provide structure to the behavior of a graph, showing trends as \( x \) heads to infinity or negative infinity.
  They don't limit the domain but frame how the graph tends to behave at extreme values.

Intercepts

The **intercepts** of a function are the points where the graph touches or crosses the axes. These provide useful landmarks for sketching out how the function sits in the coordinate space.
For the function \( y = 3 + \frac{1}{x} \):

• Y-intercept: Normally found by setting \( x = 0 \). However, as the function is undefined at \( x = 0 \), here, there isn't a y-intercept.
• X-intercept: Determined by setting \( y = 0 \). Solving \( 3 + \frac{1}{x} = 0 \) gives \( x = -\frac{1}{3} \). So, the point \(( -\frac{1}{3}, 0)\) is our x-intercept, showing where the graph crosses the x-axis.
  These intercepts enable us to accurately plot the graph initially, giving concrete starting points where the function behaves predictably.