Problem 9 - Calculus for Biology and Medicine

Question

sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=\frac{x}{x+1} $$

Step-by-Step Solution

Answer

The function has a vertical asymptote at \(x=-1\), a horizontal asymptote at \(y=1\), and an intercept at \((0,0)\).

Step 1: Identify the Domain

To find where the function is undefined, set the denominator equal to zero: \(x+1=0\). Solving gives \(x = -1\). Therefore, the domain is all real numbers except \(x = -1\).

Step 2: Determine Vertical Asymptote

The vertical asymptote occurs where the function is undefined. Since the function is undefined at \(x = -1\), there is a vertical asymptote at \(x = -1\).

Step 3: Find Horizontal Asymptote

For rational functions like \(y = \frac{x}{x+1}\), if the degrees of the numerator and the denominator are the same, the horizontal asymptote is the ratio of their leading coefficients. Here, both numerator and denominator are degree 1, so the asymptote is \(y = \frac{1}{1} = 1\).

Step 4: Calculate Intercepts

For the y-intercept, set \(x = 0\): \(y = \frac{0}{0+1} = 0\). Thus, the y-intercept is \((0, 0)\). For the x-intercept, set \(y = 0\): \(\frac{x}{x+1} = 0\) implies \(x = 0\). Thus, the x-intercept is also \((0, 0)\).

Step 5: Sketch the Graph

Plot the intercept \((0,0)\) and draw the asymptotes: vertical at \(x = -1\) and horizontal at \(y = 1\). Since the function is defined everywhere except at the vertical asymptote, sketch the curve approaching the asymptotes, increasing from \(-\infty\) on the left of \(x=-1\) and decreasing to \(1\) as \(x\) approaches \(\infty\).

Key Concepts

Asymptotes

One of the key features when sketching graphs of rational functions is identifying asymptotes. Asymptotes are lines that the graph of a function approaches but never actually touches. They are crucial for understanding the behavior of the graph at extreme ends of the x-axis and y-axis.

For the given function, \(y = \frac{x}{x+1}\), let's focus on two types of asymptotes:

• Vertical Asymptote: The vertical asymptote is determined by identifying the values of \(x\) that make the denominator zero, because division by zero is undefined. For \(y = \frac{x}{x+1}\), set the denominator \(x+1 = 0\), which implies \(x = -1\). Therefore, the vertical asymptote is the line \(x = -1\).


• Horizontal Asymptote: For rational functions where the degrees of the numerator and the denominator are the same, the horizontal asymptote is the line where \(y = \frac{a}{b}\), where \(a\) and \(b\) are the leading coefficients. Here, both numerator and denominator have degree 1 and leading coefficients of 1, giving us the horizontal asymptote \(y = 1\).

These asymptotes help us visually constrain the behavior of the function and guide us to correctly sketch the graph.

Domain of a Function

Understanding the domain of a function is essential to knowing where the function is valid and can be graphically represented. The domain consists of all the possible x-values that can be plugged into the function without making it invalid, like causing division by zero or yielding a negative square root when only real numbers are considered.

For the rational function \(y = \frac{x}{x + 1}\):

• Determine where it's undefined: Set the denominator equal to zero, \(x+1 = 0\), which results in \(x = -1\). This means the function is undefined at \(x = -1\).


• Write the domain: Exclude this value from the domain. So, the domain consists of all real numbers except \(x = -1\). Mathematically, this can be expressed as \(x \in \mathbb{R}, x eq -1\).

Understanding the domain is critical for graphing as it tells us exactly where the function exists and where asymptotes may be found.

Intercepts

Finding intercepts gives valuable anchor points where a graph crosses the axes, which can simplify the graphing process. There are two types of intercepts in graph sketching: x-intercepts and y-intercepts.

Let's find them for the function \(y = \frac{x}{x + 1}\):

• Y-Intercept: To find the y-intercept, substitute \(x = 0\) into the function. Doing this gives \(y = \frac{0}{0 + 1} = 0\). Therefore, the y-intercept is the point \((0, 0)\).


• X-Intercept: To locate the x-intercept, set \(y = 0\) and solve for \(x\). Given \(y = \frac{x}{x + 1} = 0\), the numerator must be zero, so \(x = 0\). This means the x-intercept is \((0, 0)\) as well.

Identifying intercepts is particularly useful as these points tell us where the graph crosses the axes and they provide initial points to begin sketching the overall shape of the graph.