Problem 51 - College Algebra with Corequisite Support

Question

For the following exercises, find the inverse of the functions with \(a, b, c\) positive real numbers. $$ f(x)=a x^{3}+b $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \sqrt[3]{\frac{x-b}{a}} \).

Step 1: Definition of the Problem

We are given the function \( f(x) = ax^{3} + b \) and need to find its inverse, denoted as \( f^{-1}(x) \).

Step 2: Setting Up the Equation

To find the inverse, swap \( x \) and \( y \) and solve for \( y \). Here \( y = f(x) \), so the equation becomes \( x = ay^{3} + b \).

Step 3: Solving for y

Isolate \( y^3 \) by subtracting \( b \) from both sides, which gives \( x - b = ay^3 \). Then, divide both sides by \( a \) to get \( y^3 = \frac{x-b}{a} \).

Step 4: Finding the Inverse

Take the cube root of both sides to solve for \( y \). Thus, \( y = \sqrt[3]{\frac{x-b}{a}} \). This is the expression for the inverse function, \( f^{-1}(x) = \sqrt[3]{\frac{x-b}{a}} \).

Step 5: Verifying the Inverse

To confirm, substitute \( f^{-1}(x) \) into \( f(x) \) and check if it outputs \( x \). Starting with \( f(f^{-1}(x)) = a(\sqrt[3]{\frac{x-b}{a}})^3 + b = x-b + b = x \), which verifies the inverse is correct.

Key Concepts

Function Composition

Function composition is the process of applying one function to the results of another function. In mathematical terms, if you have two functions, say \( f(x) \) and \( g(x) \), their composition is expressed as \( (f \circ g)(x) = f(g(x)) \). This means you first apply the function \( g \) to \( x \) and then apply the function \( f \) to the result of \( g(x) \).
A key concept to understand when working with inverse functions is using function composition to verify the correctness of an inverse. When you find an inverse function, \( f^{-1}(x) \), you can confirm it by checking that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).
This operation should return the identity function, which simply outputs the input \( x \) without alteration. Such verification ensures that the inverse function truly undoes the effect of the original function.

Cube Root

The cube root is a mathematical operation that finds a number which, when multiplied by itself three times, equals the given number. If you have a number \( y \), and if \( y = x^3 \), then \( x \) is the cube root of \( y \). It is denoted as \( \sqrt[3]{y} \).
In algebra, finding a cube root is the opposite operation to cubing a number. When dealing with inverse functions, especially those involving cubic terms, you often use the cube root to isolate variables. For example, in the inverse function \( f^{-1}(x) = \sqrt[3]{\frac{x-b}{a}} \), the cube root helps to solve for \( y \) by removing the cube from \( y^3 \).
Understanding cube roots make it easier to handle functions that involve higher degrees, as it allows us to revert changes caused by cubing in the original function.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying algebraic expressions using various algebraic rules and properties. This can include operations such as factoring, expanding, simplifying, and solving equations.
In the context of finding inverse functions, algebraic manipulation is crucial. You need to skillfully rearrange the original function equation to express the dependent variable (often \( y \)) in terms of the independent variable (\( x \)).
For example, to find the inverse of \( f(x) = ax^3 + b \), you manipulate the equation to get \( y^3 \) by subtracting \( b \) from both sides and then dividing by \( a \), leading to \( y^3 = \frac{x-b}{a} \).
Finally, by taking the cube root, we isolate \( y \). Algebraic manipulation is essential for effectively handling equations and ensuring you can solve for the desired variable efficiently.