Problem 38 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=\left(x^{3}+1\right)^{5} $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(y) = (y^{1/5} - 1)^{1/3} \).

Step 1: Define the Function and Its Inverse

Let's start by assuming that the function is expressed as \( y = f(x) = (x^3 + 1)^5 \). To find the inverse of this function, we need to solve for \( x \) in terms of \( y \). First, we express the function as \( x \) using \( y \).

Step 2: Express in Terms of y

Set \( y = (x^3 + 1)^5 \). To begin isolating \( x \), take the fifth root of both sides. This gives us \( (x^3 + 1) = y^{1/5} \).

Step 3: Solve for x in Terms of y

Subtract 1 from both sides: \( x^3 = y^{1/5} - 1 \). Next, take the cube root of both sides to solve for \( x \): \( x = (y^{1/5} - 1)^{1/3} \).

Step 4: Express the Inverse Function

The inverse function \( f^{-1}(y) \) is found as \( f^{-1}(y) = (y^{1/5} - 1)^{1/3} \). This inverse takes any output \( y \) from the original function and returns the original input \( x \).

Key Concepts

Function Composition

Function composition involves combining two or more functions to create a new function. You could think of it as taking the output of one function and using it as the input for another. It's like connecting pipes, where the output of one flows into the next. In mathematics, this is expressed as a composition such as \( (f \circ g)(x) \), which means you first apply \( g \) to \( x \), and then apply \( f \) to the result.

• This concept is essential when dealing with inverse functions. You are finding a function that, when composed with the original, returns the identity function \( x \).
• In the exercise, we effectively composed backward by decomposing \( f(x) \). The inverse unravels \( f \) by opposing its transformations, peeling the function layer by layer.

This backward composition is necessary to trace back from the output \( y \) to input \( x \). Understanding how inputs and outputs are interrelated helps solidify the understanding of inverse functions.

Function Transformation

Function transformation modifies a given function in a systematic way to change its graph. There are several transformations that can be applied, such as shifts, reflections, stretches, and compressions. In the given exercise, you can observe transformations through the operations applied to \( x \).

• Basic transformations include translations, where a constant is added or subtracted from \( x \), as seen in \( x^3 + 1 \).
• Power transformations alter a function by raising it to a power. In our case, \( (x^3 + 1)^5 \) is a power transformation which stretches or compresses the function.
• Finding the inverse involves reversing these transformations sequentially, effectively 'undoing' them to retrieve the original function \( x \).

Transformation is central in finding an inverse because you have to trace backwards through each transformation step to express \( x \) as a function of \( y \), resulting in the inverse \( f^{-1}(y) \).

Algebraic Manipulation

Algebraic manipulation involves using algebraic rules and operations to rearrange and solve equations. It's a core skill for working with functions, including finding inverses. Let's break down the manipulation steps used in our exercise:

• Firstly, we expressed the function in terms of \( y \), \( y = (x^3 + 1)^5 \), inverting the operations starting with taking the fifth root to isolate \( (x^3 + 1) \).
• By subtracting 1 from both sides, we then simplify the equation to \( x^3 = y^{1/5} - 1 \).
• Taking the cube root finally isolates \( x \) as \( x = (y^{1/5} - 1)^{1/3} \), showcasing a set of algebraic manipulations to solve for \( x \).

Mastery of algebraic manipulation allows you to reverse-engineer functions and find their inverses effectively. It's about knowing how to rearrange and solve equations to reveal hidden relationships between variables.