Problem 44 - Intermediate Algebra

Question

Each of the following functions is one-to-one. Find the inverse of each function and express it using \(f^{-1}(x)\) notation. $$ f(x)=\frac{1}{x} $$

Step-by-Step Solution

Answer

The inverse of the function is \( f^{-1}(x) = \frac{1}{x} \).

Step 1: Understand the Function

We have the function \( f(x) = \frac{1}{x} \). This is a basic rational function where the output is the reciprocal of the input.

Step 2: Set the Function Equal to y

Since we want to find the inverse, we start by writing the function as \( y = \frac{1}{x} \).

Step 3: Swap x and y

To find the inverse, swap \( x \) and \( y \). This gives us \( x = \frac{1}{y} \).

Step 4: Solve for y

Solve the equation \( x = \frac{1}{y} \) for \( y \). Multiply both sides by \( y \) and divide by \( x \) to get \( y = \frac{1}{x} \).

Step 5: Write the Inverse Function

Express the inverse using \( f^{-1}(x) \) notation. Thus, the inverse function is \( f^{-1}(x) = \frac{1}{x} \).

Key Concepts

Understanding One-to-One Functions

One-to-one functions are foundational in mathematics, especially when discussing inverses. A function is considered one-to-one if it assigns distinct outputs to distinct inputs. In simpler terms, every input has a unique output. This uniqueness is crucial since it ensures that the function has an inverse.
To check if a function is one-to-one, you can use the horizontal line test. This involves drawing horizontal lines across the graph of the function. If any horizontal line intersects the graph more than once, the function is not one-to-one.
Examples of one-to-one functions include linear functions with a non-zero slope and exponential functions. Understanding these functions helps with finding inverses, a critical concept in various fields of math and science.

Exploring Rational Functions

Rational functions are another important category in algebra. They are expressions that can be written as the ratio of two polynomials. That means:

• The numerator and the denominator are both polynomial functions.
• It's often expressed in the form \( f(x) = \frac{p(x)}{q(x)} \) where \( p(x) \) and \( q(x) \) are polynomials and \( q(x) eq 0 \).

Rational functions can exhibit a variety of behaviors and complexities. They can have vertical asymptotes, which happen where the denominator is zero and the function "blows up" to infinity. They can also have horizontal or oblique asymptotes.
This complexity offers a rich ground for exploring behaviors, such as finding inverses. Not all rational functions have inverses, but those that are one-to-one do. For instance, the function \( f(x) = \frac{1}{x} \) is both rational and one-to-one, allowing us to find its inverse.

Steps to Finding Inverses

Finding the inverse of a function involves a few straightforward steps:

• Express the Function as an Equation: Start by rewriting the function in the form \( y = f(x) \). For example, if \( f(x) = \frac{1}{x} \), then it becomes \( y = \frac{1}{x} \).
• Swap the Variables: Exchange the roles of \( x \) and \( y \) in the equation. Now it looks like \( x = \frac{1}{y} \).
• Solve for the New \( y \): Manipulate the equation to get \( y \) alone on one side. For our example, you multiply both sides by \( y \) and then divide by \( x \), resulting in \( y = \frac{1}{x} \).
• Express the Inverse: The result is your inverse function, written in \( f^{-1}(x) \) notation. In this case, \( f^{-1}(x) = \frac{1}{x} \).

Each step is methodical and ensures that the inverse retains the unique mapping—each input produces a distinct output. Understand these steps clearly as they are vital when working with more complex functions.