Problem 50 - 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. $$ f(x)=(x-9)^{3} $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = x^{1/3} + 9 \).

Step 1: Set the Function Equal to y

Assign the function to a variable \( y = f(x) \). Here, we start with \( y = (x-9)^3 \).

Step 2: Solve for x in Terms of y

To find the inverse, we need to solve for \( x \) in terms of \( y \). First, we take the cube root of both sides: \[ y^{1/3} = x - 9 \]. Then, solve for \( x \): \[ x = y^{1/3} + 9 \].

Step 3: Swap Variables and Write the Inverse

Rewrite the expression with \( x \) in place of \( y \), so the inverse function \( f^{-1}(x) \) becomes \( f^{-1}(x) = x^{1/3} + 9 \).

Step 4: Confirm the Inverse Function

Verify that interchanging \( x \) and \( y \) leads back to the original function, meaning if \( f(x) = (x-9)^3 \) then \( f^{-1}(x) = x^{1/3} + 9 \) correctly inverts it.

Key Concepts

One-to-One Functions

Understanding one-to-one functions is crucial for finding inverse functions. But what exactly does "one-to-one" mean? In simple terms, a one-to-one function is a type of function where each output value is mapped to by exactly one input value. No two different inputs can produce the same output. This property is very important because it ensures that the inverse of the function will be a function itself. Think of it like a lock-and-key system; each key fits exactly one lock.
Here are a few important characteristics of one-to-one functions:

• Every y-value is unique to each x-value.
• If the function is graphed, it passes the horizontal line test, meaning no horizontal line cuts the graph at more than one point.

Why do we care about one-to-one functions when finding inverses? Simply put, if a function is not one-to-one, attempting to find an inverse could result in more than one output for an input, violating the definition of a function. So always check for this property before trying to derive an inverse.

Solving Equations

Solving equations is a step-by-step method used in finding inverses, among many other applications. As seen in our exercise, solving equations helps us transition from the given function to its inverse.
Let’s break down the steps we used in our original solution. Starting with the function given as:

• Assign the function to a variable: Start by letting the function be equal to \( y \), i.e., \( y = (x-9)^3 \).
• Solve for \( x \): To find the inverse, express \( x \) in terms of \( y \). First, isolate \( x \) on one side of the equation. For example, find \( x \) by taking the cube root of both sides following up with some basic algebraic manipulations to get \( x = y^{1/3} + 9 \).

The aim here is to express \( x \) in terms of \( y \), which can be swapped in forthcoming steps to get the inverse function.

Function Notation

Function notation plays a vital role in simplifying expressions and communicating ideas in algebra. When dealing with inverse functions, we usually employ the notation \( f^{-1}(x) \) to label the inverse of \( f(x) \).
Here's a bit more on how this notation works:

• The function \( f(x) \) typically represents the original function. In our case, \( f(x) = (x-9)^3 \).
• The inverse function, represented as \( f^{-1}(x) \), is the function that 'undoes' the action of \( f(x) \). For our function, the inverse is \( f^{-1}(x) = x^{1/3} + 9 \).
• Using precise notation aids in clearly specifying which function is being referred to, especially when several functions are in play.

In practice, this notation helps avoid confusion and indicates clearly that functions and their inverses are distinct, yet interconnected.