Problem 51 - College Algebra

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

Step 1: Express Function Equation

We are provided with the function \( f(x) = ax^3 + b \). To find its inverse, we first express the function as \( y = ax^3 + b \). This will make it easier to work with.

Step 2: Solve for x

We need to solve for \( x \) in terms of \( y \). Start by subtracting \( b \) from both sides of the equation: \( y - b = ax^3 \). Next, divide both sides by \( a \) to isolate \( x^3 \): \( \frac{y - b}{a} = x^3 \).

Step 3: Take the Cube Root

To solve for \( x \), take the cube root of both sides: \( x = \sqrt[3]{\frac{y - b}{a}} \).

Step 4: Write the Inverse Function

Now that we have \( x \) in terms of \( y \), we can write the inverse function \( f^{-1}(y) = \sqrt[3]{\frac{y - b}{a}} \). To match typical function notation, we replace \( y \) with \( x \): \( f^{-1}(x) = \sqrt[3]{\frac{x - b}{a}} \).

Key Concepts

Cube Root

The cube root is a fundamental concept in mathematics, especially when dealing with inverse functions. When we have a number expressed as the cube of another, like \(x^3\), finding the cube root is about identifying what number, when raised to the power of three, will return the original value. In mathematical terms, the cube root of a number \(n\) is denoted as \(\sqrt[3]{n}\). This is different from a square root, which is denoted as \(\sqrt{}\), because instead of taking a square or the power of two, you're taking a cube or the power of three.
When working through inverse functions, if you have an expression like \(ax^3\), you solve for \(x\) by isolating \(x^3\) and then applying the cube root to find \(x\). For instance, with the equation \(x = \sqrt[3]{\frac{y - b}{a}}\), you've isolated \(x^3\) and thus take the cube root to solve for \(x\).
Understanding cube roots is essential in algebra andinverse functions because it helps unravel cubic equations, allowing you to express one variable in terms of another, and that forms the inverse function.

Algebraic Manipulation

Algebraic manipulation is a crucial skill in solving equations and finding inverse functions. It involves rearranging and simplifying algebraic expressions to solve for a particular variable.

• First, you start by setting up the equation correctly. For inverse functions, this often involves swapping \(x\) and \(y\) to solve for \(x\).
• Then, you perform operations to isolate the desired variable. This might include addition or subtraction to move terms, division or multiplication to solve for coefficients, and, as in our case, taking roots to resolve powers.

In our specific example of the function \(f(x) = ax^3 + b\), the algebraic manipulation involves:
- Subtracting \(b\) from both sides to get \(y - b = ax^3\).
- Dividing by \(a\) so \(x^3\) is isolated as \(\frac{y - b}{a} = x^3\).
This step-by-step process is essential for accurately handling equations to find the inverse functions, allowing one to express inputs and outputs in terms of one another.

Function Notation

Function notation is a standardized way to represent functions, making it easier to deal with more complex equations and transformations, such as finding inverses.
A function \(f(x)\) not only provides a mapping from an input \(x\) to an output \(f(x)\) but also lends itself to being manipulated algebraically.
When finding inverses, we often start with \(y = f(x)\), replace \(f(x)\) with \(y\), and then solve for \(x\) in terms of \(y\). After finding the expression for the inverse, we return to function notation, using \(f^{-1}(x)\) to denote the inverse function.

• In our given example, the function \(f(x) = ax^3 + b\) undergoes these transformations through algebraic manipulation.
• The inverse of the function, in function notation, is then written as \(f^{-1}(x) = \sqrt[3]{\frac{x - b}{a}}\), showing this reverse mapping from output \(f(x)\) back to the original input \(x\).

The systematic use of function notation is important for clarity and consistency, guiding us to interpret and solve more complex mathematical problems.