Problem 44 - College Algebra with Corequisite Support

Question

For the following exercises, find the inverse function. Then, graph the function and its inverse. Find the inverse function of \(f(x)=\frac{1}{x-1} .\) Use a graphing utility to find its domain and range. Write the domain and range in interval notation.

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \frac{1 + x}{x}\). The domain is \((-\infty, 0) \cup (0, \infty)\) and range is \((-\infty, 1) \cup (1, \infty)\).

Step 1: Set Up the Equation for Inversion

We begin by setting the function equal to a new variable, typically using y in place of \(f(x)\). Thus, the equation becomes: \(y = \frac{1}{x-1}\). Our goal is to find \(x\) in terms of \(y\).

Step 2: Solve for x

To find the inverse, solve the equation \(y = \frac{1}{x-1}\) for \(x\). Multiply both sides by \(x-1\) to get \(y(x-1) = 1\). Simplify to \(yx - y = 1\). Solve for \(x\) by adding \(y\) to both sides and dividing by \(y\): \(x = \frac{1 + y}{y}\).

Step 3: Express the Inverse Function

Replace \(y\) with \(x\) in the equation \(x = \frac{1 + y}{y}\) to express the inverse function. The inverse function is \(f^{-1}(x) = \frac{1+x}{x}\).

Step 4: Determine Domain and Range of the Original Function

For \(f(x) = \frac{1}{x-1}\), the domain is all real numbers except the value that makes the denominator zero. Thus, Domain = \((-\infty, 1) \cup (1, \infty)\). The range is all real values except zero, so Range = \((-\infty, 0) \cup (0, \infty)\).

Step 5: Determine Domain and Range of the Inverse Function

For the inverse function \(f^{-1}(x) = \frac{1 + x}{x}\), the domain is all real numbers except zero, Domain = \((-\infty, 0) \cup (0, \infty)\). The range is all real numbers except 1, Range = \((-\infty, 1) \cup (1, \infty)\).

Step 6: Verify by Graphing

Graph both the original function \(f(x) = \frac{1}{x-1}\) and its inverse \(f^{-1}(x) = \frac{1+x}{x}\). The graphs of a function and its inverse should be reflections of each other across the line \(y = x\). Use graphing software or a graphing calculator to confirm the graph and observe the domain and range from the graph.

Key Concepts

Domain and Range

Understanding the domain and range of a function is crucial when we are dealing with inverse functions. In the original function, domain refers to all possible input values, while range refers to all potential output values.
For the function \(f(x) = \frac{1}{x-1}\), the domain excludes \(x = 1\) because it makes the denominator zero, leading to undefined output. Thus, the domain is

• Domain: \((-\infty, 1) \cup (1, \infty)\)

For the range, any non-zero real number is possible because the fraction can take any positive or negative value other than zero. Therefore, the range is

• Range: \((-\infty, 0) \cup (0, \infty)\)

In contrast, the inverse function \(f^{-1}(x) = \frac{1+x}{x}\) has a domain that excludes \(x = 0\) to avoid division by zero:

• Domain: \((-\infty, 0) \cup (0, \infty)\)

And the range is all real numbers except 1, since the expression \(\frac{1+x}{x}\) will never equal 1 for any real \(x\):

• Range: \((-\infty, 1) \cup (1, \infty)\)

Graphing Utilities

Graphing utilities like graphing calculators or graphing software are powerful tools to visualize functions and their inverses. By graphing \(f(x) = \frac{1}{x-1}\) and its inverse \(f^{-1}(x) = \frac{1+x}{x}\), you can see their symmetry around the line \(y = x\).
This line is a visual guide that helps to confirm the correctness of inverse functions. When you plot both functions, they should mirror each other across this line.

• Visual checking: The inverse \(f^{-1}(x)\) will swap the x-axis with the y-axis compared to \(f(x)\).
• Functional understanding: Ensure any transformation or manipulation of the graph doesn’t violate the properties of reflection over \(y = x\).

Using different zoom levels or tracing features of graphing software can provide insights into subtle changes in domain and range, which are visible on the graph as open or filled dots indicating non-inclusivity or inclusivity of points.

Algebraic Expressions

Manipulating algebraic expressions is a cornerstone of finding inverse functions. This involves solving for \(x\) in terms of \(y\) and then swapping the variables. Let's break this down:

For the given function \(y = \frac{1}{x-1}\), we first multiply both sides by \(x-1\) to move the expression in the denominator to the numerator on one side:

• Multiply: \(y(x-1) = 1\).
• Simplify: Expand to \(yx - y = 1\).
• Solve for \(x\): Rearrange to get \(x = \frac{1+y}{y}\).

Once you derive \(x\) in terms of \(y\), substitute \(y\) with \(x\) to find the inverse function \(f^{-1}(x)\). This manipulation with algebraic expressions allows us to express complexity in a manageable form, providing insights into the flip of domain and range between a function and its inverse. Understanding these manipulations is vital for function inversions and transformations. The practice of careful algebraic manipulation helps build intuition for the interchanges happening during such processes.