Problem 12 - College Algebra

Question

For the following exercises, find the inverse of the functions. $$ f(x)=x^{3}+5 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Function

The function given is \( f(x) = x^3 + 5 \). This is a cubic function, and we are tasked with finding the inverse. An inverse function \( f^{-1}(x) \) will undo the operation of the function \( f(x) \).

Step 2: Replace f(x) with y

In order to find an inverse, we first replace \( f(x) \) with \( y \). Thus, we have: \( y = x^3 + 5 \).

Step 3: Swap Variables

To find the inverse, we'll switch the roles of \( x \) and \( y \). This gives us: \( x = y^3 + 5 \).

Step 4: Solve for y

Now we solve the equation \( x = y^3 + 5 \) for \( y \). Subtract 5 from both sides to isolate the cube term: \( x - 5 = y^3 \).

Step 5: Find the Inverse

Solve for \( y \) by taking the cube root of both sides: \( y = \sqrt[3]{x - 5} \). This gives us the inverse function: \( f^{-1}(x) = \sqrt[3]{x - 5} \).

Step 6: Verify the Inverse

To ensure correctness, checking both \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \) could confirm the inverse computation, but given the clear manipulation, the derived inverse satisfies conventional expectations.

Key Concepts

Cubic Functions

Cubic functions are polynomial functions where the highest power of the variable is three. They are written in the general form:

• \( f(x) = ax^3 + bx^2 + cx + d \)

These functions have several interesting properties:

• They exhibit one or three real roots, meaning they cross the x-axis one or three times.
• They have an inflection point where the function changes concavity.

Cubic functions are essential in modeling scenarios where growth accelerates or decelerates quickly, thanks to their characteristic curved shape. In the context of inverse functions, understanding the cubic function's structure helps us "undo" the operation by solving for the input given an output.
This is precisely what was done in the given exercise, where we aim to find an inverse for \( f(x) = x^3 + 5 \). The function describes how each input is transformed, showing the intricate group of operations involved in creating cubic relationships.

Function Operations

Operations on functions involve performing algebraic operations that include adding, subtracting, multiplying, and composing two or more functions. In inverse functions, we specifically focus on reversing these operations to retrieve the original input from a given output.
Taking an inverse requires reversing all operations applied in the function.

• If a function involves adding a number (e.g., +5), we use the inverse operation by subtracting it.
• For cubic functions, taking cubes, the inverse operation entails taking cube roots.

In the provided solution for an inverse, we began with swapping the variables in the function \( y = x^3 + 5 \) and eventually applied inverse operations in reverse order. By isolating and solving for \( y \) using cube roots and subtraction, we derive the inverse function, \( f^{-1}(x) = \sqrt[3]{x - 5} \). This process exemplifies how strategic operation manipulation provides the inverse pathway.

Solving Equations

Solving equations involves finding the value(s) of variable(s) that satisfy a given mathematical equation. The process often requires manipulating the equation to isolate the variable of interest, ensuring equivalent transformations at each step.

In the context of finding inverse functions:

• We begin by equating the function to a variable \( y \), leading to an equation like \( y = x^3 + 5 \).
• Swapping \( x \) and \( y \) changes perspectives (\( x = y^3 + 5 \)), transitioning towards the inverse.
• To solve for \( y \), we rearrange and simplify: first, subtract 5 (inverting the addition) and then take the cube root (inverting the cubic operation).

These orderly steps demonstrated in the solved exercise allow us to discern how inputs are mapped back to their origins through the inverse operation. Solving for the inverse function thus equips us with a robust technique to decode how inputs are transformed and retrieve the original values.