Problem 19 - College Algebra

Question

The functions in Exercises \(11-30\) are all one-to-one. For each function: a. Find an equation for \(f^{-1}(x),\) the inverse function. b. Verify that your equation is correct by showing that $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=(x+2)^{3}$$

Step-by-Step Solution

Answer

The inverse function of \(f(x) = (x + 2)^{3}\) is \(f^{-1}(x) = \sqrt[3]{x} - 2\).

Step 1: Finding the Inverse Function

The given function is \(f(x)=(x+2)^3\). In order to find its inverse, replace \(f(x)\) with \(y\). So, the equation becomes \(y=(x+2)^3\). Next, interchange the roles of y and x. This gives \(x=(y+2)^3\). Now, solve for y: take the cube root of both sides \(y=\sqrt[3]{x}-2\). This is the inverse function, denoted by \(f^{-1}(x)=\sqrt[3]{x}-2\).

Step 2: Verifying the Inverse Function

Verification is done by showing that \(f(f^{-1}(x))=x\) and \(f^{-1}(f(x))=x\). Substituting \(f^{-1}(x)\) into \(f(x)\) we have \(f(f^{-1}(x))=((\sqrt[3]{x}-2)+2)^3=x\). Which simplifies to \(x\). Substituting \(f(x)\) into \(f^{-1}(x)\), we have \(f^{-1}(f(x))=\sqrt[3]{((x+2)^3)}-2\). Which simplifies to \(x\). Thus, as both the conditions are satisfied, the derived inverse function is correct.

Key Concepts

One-to-One Function

A one-to-one function is a type of function where every element in the domain is paired with a unique element in the codomain. This means that if you pick any two different inputs, they will always have different outputs. For a function to be one-to-one, each x-value must correspond to exactly one y-value, and no y-value can be assigned to more than one x-value.

It is important to know if a function is one-to-one because it directly determines if the function has an inverse. If a function is not one-to-one, its inverse either doesn't exist or isn't a true function. In the exercise, we're told that the function is one-to-one, so we can proceed with finding its inverse confidently.

Function Verification

Function verification ensures that the derived inverse is correct. For a function and its inverse, two important conditions need to be satisfied:

• First, when you substitute the inverse of a function into the original function, you should get back the original input value. Mathematically, it is expressed as \(f(f^{-1}(x)) = x\).
• Second, when you substitute the original function into its inverse, it should also return to the original input value. This is expressed as \(f^{-1}(f(x)) = x\).

Verifying the function by these conditions confirms whether the inversion calculations are performed correctly. In our exercise, as shown in the solution, both conditions hold true. By performing these steps in verification, confidence in the inverse function’s accuracy is boosted.

Cube Root

Cube roots are the inverse process of cubing a number. If you have a number, say \(a\), the cube root of \(a\) is a number \(b\) such that \(b^3 = a\). When working with cube roots in mathematics, they help to "reverse" the action of cubing a number, which is raising to the power of three.

In the context of finding inverse functions, such as in our exercise, taking the cube root was essential to undo the cubing in the original function \(f(x) = (x+2)^3\). By applying the cube root, the function was rewritten, allowing us to isolate \(x\) and express the inverse function correctly. Understanding the operation of a cube root is fundamental when manipulating equations algebraically.

Algebraic Manipulation

Algebraic manipulation involves rearranging and rewriting equations to solve for a particular variable or to simplify expressions. This process requires a firm grasp of algebraic rules and operations such as addition, subtraction, multiplication, division, and exponentiation.

In our exercise, algebraic manipulation was essential in the step where roles of \(x\) and \(y\) were interchanged, leading to \(x = (y+2)^3\). Further manipulation involved solving for \(y\) by taking the cube root and isolating \(y\) on one side of the equation. The result was the expression for the inverse function \(f^{-1}(x)= \sqrt[3]{x} - 2\).

• Understanding how to perform these operations correctly can simplify seemingly complex problems into more manageable steps.
• This skill is critical when working across various math disciplines, not just inverse functions.