Problem 39 - Calculus for Biology and Medicine

Question

Let $$ f(x)=\frac{1}{x+1}, \quad x>-1 $$ (a) Use a graphing calculator to graph \(f(x)\). (b) On the basis of the graph in (a), determine the range of \(f(x)\). (c) For which values of \(x\) is \(f(x)=2\) ? (d) On the basis of the graph in (a), determine how many solutions \(f(x)=a\) has, where \(a\) is in the range of \(f(x)\)

Step-by-Step Solution

Answer

(a) Plot shows \((-\infty,0)\cup(0,\infty)\). (b) Range is \((-\infty,0)\cup(0,\infty)\). (c) \(x = -\frac{1}{2}\). (d) One solution for every \( a \neq 0 \).

Step 1: Understand the Function

The function given is \( f(x) = \frac{1}{x+1} \). It is defined for all \( x > -1 \). This function is a simple rational function with a vertical asymptote at \( x = -1 \) where the denominator becomes zero.

Step 2: Graph the Function

Using a graphing calculator, plot the function \( f(x) = \frac{1}{x+1} \) for values of \( x \) greater than \(-1\). Observe the behavior as \( x \to -1^+ \) and as \( x \to \infty \). The graph should show an asymptote at \( x = -1 \) and approach 0 as \( x \to \infty \).

Step 3: Determine the Range from the Graph

From the graph, observe that \( f(x) \) can take any value from \( -\infty \) to \( 0 \) as \( x \to -1^+ \) and any value greater than 0. Thus, the range is \( f(x) \in (-\infty, 0) \cup (0, \infty) \).

Step 4: Solve for \( f(x) = 2 \)

Set \( f(x) = 2 \), giving the equation \( \frac{1}{x+1} = 2 \). Solve for \( x \) by multiplying each side by \( x+1 \) and rearranging: \( 1 = 2(x+1) \Rightarrow 1 = 2x + 2 \Rightarrow 2x = -1 \Rightarrow x = -\frac{1}{2} \).

Step 5: Infer Solutions for \( f(x) = a \)

From the graph, the function \( f(x) \) has one horizontal asymptote at 0. For any \( a \in (-\infty, 0) \cup (0, \infty) \), the function \( f(x) \) will have exactly one solution since \( f(x) \) is continuous and one-to-one in each partition of its range.

Key Concepts

Asymptotes

In the context of rational functions, asymptotes are lines that the graph of a function approaches but never actually touches. There are two main types: **vertical** and **horizontal** asymptotes. Vertical asymptotes occur where the denominator of a rational function is zero and the function is undefined, resulting in a value that trends towards infinity or negative infinity. For the function \( f(x) = \frac{1}{x+1} \), the denominator \( x+1 \) becomes zero at \( x = -1 \). Hence, there is a vertical asymptote at \( x = -1 \).Horizontal asymptotes describe the behavior of a function as \( x \) approaches positive or negative infinity. In our function, as \( x \to \infty \), \( f(x) \to 0 \). Therefore, the horizontal asymptote is the line \( y = 0 \). The graph gets closer and closer to this line, but never actually reaches it.

Graphing Functions

Graphing rational functions, like \( f(x) = \frac{1}{x+1} \), provides visual insights into their behavior. When graphing such functions with a graphing calculator, you can observe key elements like asymptotic behavior and domain and range limitations. **Steps to Graph**:

• Identify the asymptotes. For \( f(x) \), the vertical asymptote is \( x = -1 \) and the horizontal asymptote is \( y = 0 \).
• Plot some points as \( x \) approaches the vertical asymptote from the right with values such as \( x=0 \), \( x=1 \), etc.
• Observe the curve as it approaches infinite values near its asymptotes.

Watching the function behavior as \( x \) nears the vertical asymptote and as \( x \) approaches infinity, helps to fully understand how the function behaves across its domain.

Solving Equations

To solve equations involving rational functions, like \( f(x) = 2 \), you'll need to set the function equal to the desired value and solve for \( x \). This process often involves isolating \( x \) by handling the equation algebraically. For \( f(x) = \frac{1}{x+1} \) when \( f(x) = 2 \):

• Set the equation: \( \frac{1}{x+1} = 2 \).
• Clear the fraction by multiplying by \( x+1 \), leading to: \( 1 = 2(x+1) \).
• Simplify and solve for \( x \): \( 1 = 2x + 2 \Rightarrow 2x = -1 \Rightarrow x = -\frac{1}{2} \).

Each of these steps helps break down the problem, making it easier to find where the function equals a specific value.

Function Range

The range of a function describes all the possible outputs (\( y \)-values) it can take. For a rational function like \( f(x) = \frac{1}{x+1} \), understanding the range requires observing the graph and determining which values \( f(x) \) can achieve. From the graph, \( f(x) \) can take values less than 0 and greater than 0, as it trends towards \( negative\) and \(positive \infty \). The function never actually reaches 0, thus the range is \((-\infty, 0) \cup (0, \infty) \). This means that the graph does not cover the value 0, since it has a horizontal asymptote at \( y = 0 \), indicating \( f(x) \) will never actually reach or cross this line.