Problem 54 - 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. \(f(x)=\frac{x^{9}}{4}\)

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \sqrt[9]{4x}\).

Step 1: Understand the Function

We are given the function \(f(x) = \frac{x^9}{4}\). This function is one-to-one, which means it has an inverse. Our goal is to find this inverse, denoted \(f^{-1}(x)\).

Step 2: Interchange x and y

First, replace \(f(x)\) with \(y\) such that \(y = \frac{x^9}{4}\). To find the inverse, we swap \(x\) and \(y\), resulting in the equation \(x = \frac{y^9}{4}\).

Step 3: Solve for y

To solve for \(y\), multiply both sides of the equation by 4 to isolate the \(y^9\) term: \(4x = y^9\).

Step 4: Simplify to Find the Inverse

To isolate \(y\), take the ninth root of both sides: \(y = \sqrt[9]{4x}\). Thus, the inverse function can be expressed as \(f^{-1}(x) = \sqrt[9]{4x}\).

Step 5: Express the Solution

The inverse function of \(f(x) = \frac{x^9}{4}\) is expressed as \(f^{-1}(x) = \sqrt[9]{4x}\) in proper notation.

Key Concepts

One-to-One Functions

A function is considered "one-to-one" if each input is paired with exactly one unique output, and each output is paired with exactly one unique input. This is a crucial characteristic for a function to have an inverse. If two different outputs come from the same input, it's not one-to-one and an inverse cannot be defined uniquely. Recognizing one-to-one functions can be achieved using several methods:

• **Horizontal Line Test**: Draw 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.
• **Checking Algebraically**: Ensure for a function \(f(x)\), if \(f(a) = f(b)\) then \(a = b\) must hold true for all input values \(a\) and \(b\).

Thorough understanding of whether a function is one-to-one or not is important. It's the first step to determine if the function can have an inverse.

Finding Inverses

To find the inverse of a function is like turning it into its opposite. This involves a systematic approach that's the same every time.Here’s a simple, step-by-step guide you can follow:

• **Replace**: Start by switching the function notation \(f(x)\) with \(y\). It's easier when you work with an equation.
• **Swap Variables**: Exchange every instance of \(x\) with \(y\) and \(y\) with \(x\). This gives you the equation you need for finding the inverse.
• **Solve for \(y\)**: Rearrange the equation to solve for \(y\). This step will vary substantially depending on the original function.
• **Finalize**: Once \(y\) is isolated, replace it with the inverse function notation \(f^{-1}(x)\).

It’s important to note that not all functions will have inverses, only those with a one-to-one correspondence between inputs and outputs.

Notation for Inverse Functions

In mathematics, notations simplify the way we express complex ideas and processes. When dealing with inverse functions, the notation \(f^{-1}(x)\) is used to denote the inverse of a function \(f(x)\).Here are some key points about this notation:

• **Inverse Does Not Mean Reciprocal**: Though it might look like it, the "negative one" superscript is **not** an exponent. It does not imply a reciprocal such as \(\frac{1}{f(x)}\).
• **Function Behavior**: Using \(f^{-1}(x)\) implies that when you apply \(f\) to \(f^{-1}(x)\), you will get \(x\) back because they cancel each other out like reversing a process.
• **Using and Recognizing**: When you see \(f^{-1}(x)\), understand it signifies a mirror-like, opposite operation to \(f(x)\).

Grasping this notation is crucial. It reveals a deep and elegant feature of functions—that each inverse function acts as a 'counterbalance' to its original function, allowing you to "undo" each step taken by \(f\).