Problem 14 - Contemporary Precalculus

Question

Use algebra to find the inverse of the given one-to-one function. $$f(x)=\left(x^{5}+1\right)^{3}$$

Step-by-Step Solution

Answer

Answer: The inverse of the function is $$f^{-1}(x) = \sqrt[5]{\sqrt[3]{x} - 1}$$.

Step 1: Express the given function in terms of x and y

We are given the function $$f(x) = (x^5 + 1)^3$$. Let $$y = f(x)$$. Then our function can be written as $$y = (x^5 + 1)^3$$.

Step 2: Swap the roles of x and y

Next, we need to swap the roles of x and y in our equation. We do this by replacing x with y and y with x: $$x = (y^5 + 1)^3$$.

Step 3: Solve the new equation for y

Now, we need to solve for y in the equation $$x = (y^5 + 1)^3$$. First, take the cube root of both sides: $$\sqrt[3]{x} = y^5 + 1$$. Next, subtract 1 from both sides: $$\sqrt[3]{x} - 1 = y^5$$. Finally, take the fifth root of both sides: $$y = f^{-1}(x) = \sqrt[5]{\sqrt[3]{x} - 1}$$. So the inverse of the given one-to-one function is: $$f^{-1}(x) = \sqrt[5]{\sqrt[3]{x} - 1}$$.

Key Concepts

Understanding One-to-One Functions

A one-to-one function is a special type of function that ensures every output is paired with exactly one input. This is important for finding inverses because an inverse can only exist if each output of the original function corresponds to a unique input.
To check if a function is one-to-one, you can use the horizontal line test. If every horizontal line crosses the graph of the function at most once, then the function is one-to-one.
This characteristic allows the function to have an inverse. The inverse function essentially "reverses" the original, swapping the roles of inputs and outputs. This swap means that the inverse of a one-to-one function, if one exists, will also be a function.

Mastering Algebraic Manipulation

Algebraic manipulation is a fundamental skill in mathematics. It involves rearranging and simplifying expressions to solve equations or find the inverse of functions.
In the step-by-step solution, we used several algebraic techniques:

• Swapping variables - after expressing the function as \( y = (x^5 + 1)^3 \), we swap \( x \) and \( y \) to find the inverse equation.
• Taking roots - we took the cube root and the fifth root to isolate \( y \) and solve for it.
• Rearranging equations - using operations like subtracting 1 from both sides to proceed towards the solution.

Each of these steps is essential for progressing through an equation towards finding a solution, like isolating the output \( y \), which ultimately gives us the inverse function.

Decoding Function Notation

Function notation is a way to safely convey information about how inputs relate to outputs in a function. When dealing with inverses, using the correct notation is crucial.
The original function is expressed as \( f(x) = (x^5 + 1)^3 \). Here, \( f \) denotes the function, and \( x \) represents the input.
To find the inverse, we denote it as \( f^{-1}(x) \). This notation tells us that we're reversing the process of \( f(x) \). Instead of processing \( x \) to get an output, we input an output value to retrieve the original \( x \).
Understanding this notation helps greatly in solving problems where switching inputs and outputs is required, such as finding inverse functions.