Problem 35 - Elementary and 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. \(f(x)=2 x+4\)

Step-by-Step Solution

Answer

\( f^{-1}(x) = \frac{x - 4}{2} \)

Step 1: Understanding One-to-One Functions

A one-to-one function is a type of function where each output is uniquely related to exactly one input. This means the function has an inverse, which we can find by following a series of algebraic steps.

Step 2: Replacing f(x) with y

Begin by replacing the function notation \(f(x)\) with \(y\) to make it easier to manipulate. So, we write:\[ y = 2x + 4 \]

Step 3: Interchanging x and y

To find the inverse function, swap \(x\) and \(y\). This gives us:\[ x = 2y + 4 \]

Step 4: Solving for y

Solve the equation for \(y\) to find the inverse function:Subtract 4 from both sides:\[ x - 4 = 2y \]Divide both sides by 2:\[ y = \frac{x - 4}{2} \]

Step 5: Writing the Inverse Function

Now express \(y\) as \(f^{-1}(x)\). Thus, the inverse function is:\[ f^{-1}(x) = \frac{x - 4}{2} \]

Key Concepts

Understanding One-to-One Functions

A one-to-one function is a very special type of function. In this kind of function, each input is paired with a unique output, and vice versa. This means no two different inputs share the same output. This unique matching makes it possible to "reverse" the process through an inverse function.
Here’s a simple analogy: think of a student ID system, where each student has a unique ID number. In a one-to-one system, every student (input) corresponds to one ID (output), and each ID matches exactly one student.

• One-to-one means one input for one output.
• This property is essential for a function to have an inverse.

Understanding this concept is vital because only one-to-one functions can be inverted. Without this property, there would be ambiguity when trying to go backwards from an output to an input.

Exploring Function Notation

Function notation is a way to represent functions in mathematics, and it’s typically expressed as \(f(x)\). This notation tells us what the input is and helps us understand the relationship between input and output.
When we write \(f(x) = 2x + 4\), the \(f(x)\) symbolizes the result of the function when you put an \(x\) value into it.

• \(f(x)\) means "function of x" or "the output when x is input."
• It provides a compact way to specify the operation being conducted.

Moreover, function notation is crucial when discussing inverse functions. The inverse function uses \(f^{-1}(x)\), which indicates that you are now inputting the output from the original function to find the original input. This concise notation makes complex ideas easier to handle as we solve mathematical problems.

Steps to Finding the Inverse of a Function

Finding the inverse of a function involves a few straightforward steps. These steps reverse the operation of the original function. Let’s break it down using our example function \(f(x) = 2x + 4\).
1. **Replace \(f(x)\) with \(y\)**: Begin by substituting \(f(x)\) with \(y\) to simplify maneuvers, so you have \(y = 2x + 4\).2. **Swap \(x\) and \(y\)**: Next, exchange \(x\) and \(y\) in the equation. This gives \(x = 2y + 4\).3. **Solve for \(y\)**: Now, you rearrange the equation to solve for \(y\). Subtract 4 from both sides to get \(x - 4 = 2y\). Then divide everything by 2, resulting in \(y = \frac{x - 4}{2}\).4. **Express as \(f^{-1}(x)\)**: Finally, denote \(y\) as \(f^{-1}(x)\), indicating it’s the inverse function. Therefore, you write \(f^{-1}(x) = \frac{x - 4}{2}\).

• The function \(f\) is reversed, swapping roles of input and output.
• Each step aligns with reversing the original function's driving operations.

Following these steps consistently transforms the original function into its inverse, unveiling the function’s backward path.