Problem 19 - College Algebra

Question

The functions 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 is \( f^{-1}(x) = \sqrt[3]{x} - 2 \)

Step 1: Find the inverse function

To find the inverse, replace \( f(x) \) with \( y \). So, we have \( y = (x+2)^3 \). Then, you swap \( x \) and \( y \) to get \( x = (y+2)^3 \). Now, solve for \( y \) to get the inverse function, which results in \( y = \sqrt[3]{x} - 2 \). Therefore, \( f^{-1}(x) = \sqrt[3]{x} - 2 \)

Step 2: Substitution for Verification Part 1

To verify our answer, we have to check two identities for inverse functions - \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). Start by substituting \( f^{-1}(x) \) into \( f(x) \): \( f(f^{-1}(x)) = ((\sqrt[3]{x} - 2) + 2)^3 = x \). This verifies the first identity.

Step 3: Substitution for Verification Part 2

Now check the second identity \( f^{-1}(f(x)) = x \). Substitute \( f(x) \) into \( f^{-1}(x) \): \( f^{-1}(f(x)) = \sqrt[3]{(x+2)^3} - 2 = x \). This verifies that the proposed inverse function is indeed correct.

Key Concepts

One-to-One Functions

In mathematics, a function is one-to-one if each element of the domain maps to a unique element of the range. This means that no two different inputs will produce the same output. A visual way to understand one-to-one functions is using the horizontal line test: if any horizontal line intersects the graph of the function at most once, then the function is one-to-one.
This property is crucial when working with inverse functions. Since each input corresponds to a unique output, it allows the function to be reversed. Essentially, a one-to-one function assures us that its inverse is also a function.
For instance, in our exercise, the function \( f(x) = (x+2)^3 \) is one-to-one because it is a cubic function, and all cubic functions that include a horizontal shift and dilation are one-to-one. This characteristic ensures that its inverse \( f^{-1}(x) = \sqrt[3]{x} - 2 \) is a valid function.

Function Verification

Verifying a function, and specifically its inverse, involves confirming that composing the function with its inverse returns the original input. This is checked by two conditions:

• \( f(f^{-1}(x)) = x \)
• \( f^{-1}(f(x)) = x \)

These are the key identities of inverse functions. In our scenario, we found the inverse function as \( f^{-1}(x) = \sqrt[3]{x} - 2 \). To confirm its correctness, we substituted this into \( f(x) \) and demonstrated:
- For \( f(f^{-1}(x)) \): By substituting the inverse function into the original function, we get \( ((\sqrt[3]{x} - 2) + 2)^3 = x \). This returns the original value \( x \), thus verifying the first condition.
- For \( f^{-1}(f(x)) \): Substituting the original function \( (x+2)^3 \) into the inverse, we simplify to \( \sqrt[3]{(x+2)^3} - 2 = x \). Both identities holding true confirm that our function and its inverse are accurate.

Algebraic Manipulation

Algebraic manipulation is a mathematical technique used to rearrange and simplify equations or expressions to solve for a variable. It is fundamental when finding the inverse of a function.
In our exercise, we began with the equation \( y = (x+2)^3 \) after renaming \( f(x) \) as \( y \). To find the inverse, we performed the following algebraic steps:

• First, swap the roles of \( x \) and \( y \) to get \( x = (y+2)^3 \), preparing the equation for inversion.
• To isolate \( y \), apply the cube root to both sides, leading to \( y+2 = \sqrt[3]{x} \).
• Finally, solve for \( y \) by subtracting 2, obtaining \( y = \sqrt[3]{x} - 2 \).

These steps are organized to carefully undo the operations applied to \( x \) in the function \( f \), thereby revealing the inverse function. Using algebraic manipulation, we systematically reverse the function's operations, ensuring the inverse obtained correctly reflects the original function's transformation.