Problem 36 - 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. \(f(x)=5 x-1\)

Step-by-Step Solution

Answer

The inverse of the function is \( f^{-1}(x) = \frac{x + 1}{5} \).

Step 1: Understanding the Definition of an Inverse Function

An inverse function essentially means "undoing" the operation performed by the original function. For a function \( f(x) \), the inverse is denoted as \( f^{-1}(x) \) and satisfies the condition \( f(f^{-1}(x)) = x \). Thus, finding the inverse of a function involves reversing the operations performed.

Step 2: Set the Output of the Function Equal to y

First, express the function in terms of \( y \) instead of \( f(x) \). We write: \[ y = 5x - 1 \] This is simply renaming \( f(x) \) to \( y \) for convenience in finding the inverse.

Step 3: Solve for x in Terms of y

The next step is to express \( x \) in terms of \( y \). Start by isolating the term containing \( x \) on one side: - Add 1 to both sides: \[ y + 1 = 5x \]- Then, divide both sides by 5 to solve for \( x \): \[ x = \frac{y + 1}{5} \]

Step 4: Interchange x and y

To complete the process of finding the inverse, swap \( x \) and \( y \) in the equation derived in the previous step. This effectively gives us the inverse function:\[ y = \frac{x + 1}{5} \]

Step 5: Express the Inverse in Standard Notation

Finally, express the function you found in the standard inverse notation:\[ f^{-1}(x) = \frac{x + 1}{5} \]

Key Concepts

Understanding One-to-One Functions

A one-to-one function, or injective function, is a function where each element of the range is paired with exactly one element of the domain. This means that no two different inputs produce the same output.
Think about it as having a unique key for each lock. Every key (input) opens only one specific lock (output), and vice versa.

In the context of inverse functions, a one-to-one function is essential. Why? Because the inverse can only assign the same value from the range back to one specific input in the domain.
Here are some important points about one-to-one functions to remember:

• For every unique value of the function (output), there is exactly one unique input value (input).
• One-to-one functions always have inverses that are also functions.
• You can use the horizontal line test on the graph of a function to check if it is one-to-one: if any horizontal line intersects the graph at most once, the function is one-to-one.

Function Notation and its Importance

Function notation is a way to express relationships between elements in the domain and range clearly and concisely. Rather than using variable expressions, notation like \( f(x) \) makes it clear that we are dealing with functions.
This notation tells us we have a function named \( f \) and here's how you calculate it based on the value of \( x \).

There are several benefits to using function notation:

• Clarity: It distinguishes functions from regular equations. When you see \( f(x) \), you immediately know you're dealing with a function.
• Simplicity: Helps in expressing operations on functions, such as forming an inverse function as \( f^{-1}(x) \).
• Adaptability: Easily adjusts to operations like transformations, compositions, and inverses within mathematical expressions.

In the inverse function exercise, function notation plays a vital role. When we express the inverse function as \( f^{-1}(x) \), it indicates this new function reverses the action of \( f(x) \). This communicates the relation effectively, showing the clear link between the original function and its inverse.

Solving Equations to Find Inverse Functions

Solving equations is a fundamental skill in algebra, especially when it comes to finding inverse functions. When we solve for inverses, we're essentially reversing the steps of the initial function.
Here's a simple breakdown of how we solve equations to find inverse functions, step by step:

• **Set the Equation:** Start by letting \( y = f(x) \). This step reformulates the function as an equation where \( y \) is the output.
• **Solve for \( x \):** Manipulate the equation to express \( x \) in terms of \( y \). This involves using algebraic techniques to isolate \( x \).
• **Swap Variables:** Once you isolate \( x \), interchange \( x \) and \( y \). This step essentially defines the operation of the inverse function.
• **Use Notation:** Finally, express the resulting equation in inverse function notation as \( f^{-1}(x) \). This notation shows its role in reversing the original function \( f(x) \).

This systematic process helps to find inverse functions effectively, ensuring clear understanding of how each operation is undone by its inverse.