Problem 13 - College Algebra with Corequisite Support

Question

For the following exercises, find the inverse of the functions. $$ f(x)=3 x^{3}+1 $$

Step-by-Step Solution

Answer

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

Step 1: Replace f(x) with y

To find the inverse of the given function, we start by rewriting it as an equation with y. Replace \( f(x) \) with \( y \), so the equation becomes \( y = 3x^3 + 1 \).

Step 2: Swap x and y

Next, interchange x and y to begin solving for the inverse. This gives us \( x = 3y^3 + 1 \).

Step 3: Isolate the term with y

Subtract 1 from both sides of the equation to isolate the term with y: \( x - 1 = 3y^3 \).

Step 4: Solve for y^3

Divide both sides by 3 to solve for \( y^3 \): \( y^3 = \frac{x - 1}{3} \).

Step 5: Solve for y

Take the cube root of both sides to solve for y: \( y = \sqrt[3]{\frac{x - 1}{3}} \).

Step 6: Write the inverse function

The inverse function is found. Replace \( y \) with \( f^{-1}(x) \), so the inverse function is \( f^{-1}(x) = \sqrt[3]{\frac{x - 1}{3}} \).

Key Concepts

Function Notations

Function notation is a way to denote a function and its values. It's like a special label for functions that helps us understand and organize their use and relationships. When we see something like \( f(x) \), we know:

• "\( f \)" is the name of the function.
• "\( x \)" is the input (or variable) for this function.
• "\( f(x) \)" represents the output when we input \( x \).

Function notation is vital in mathematics because it clearly shows which variable is to be used and helps track different functions. It also comes in handy when expressing inverse functions, as seen in our example:
- The original function is \( f(x) = 3x^3 + 1 \).
- The inverse function is denoted as \( f^{-1}(x) \), meaning the function that "undoes" what \( f(x) \) does. Here it is \( f^{-1}(x) = \sqrt[3]{\frac{x - 1}{3}} \).
Understanding function notations makes it easier to work with complex functions and their inverses.

Cubic Functions

A cubic function is a type of polynomial function with the highest power of the variable being three. In our exercise, the function \( f(x) = 3x^3 + 1 \) is a cubic function. Let's break down its characteristics:

• General form: A cubic function generally looks like \( ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants, with \( a eq 0 \).
• The graph of a cubic function has an S-like shape, due to the highest power being odd.
• Cubic functions can have up to three real roots and two turning points.
• They exhibit a point of inflection where the concavity changes direction.

In our specific example, the function \( f(x) = 3x^3 + 1 \):
- Has no quadratic or linear terms, making its graph relatively simple.
- It is the type that grows quickly in both positive and negative directions from the origin.
Understanding the nature of cubic functions aids in interpreting their inverses and behavior when graphed.

Algebraic Manipulations

Algebraic manipulations are the steps needed to solve equations or transform expressions. This skill can be like solving a puzzle, ensuring each piece fits perfectly to get a clear answer. Let's look at how algebraic manipulations help find inverse functions with our example:
1. **Replacing the Notation**: We start by replacing \( f(x) \) with \( y \) to simplify manipulations.
2. **Swapping Variables**: Then, we swap \( x \) and \( y \), leading to \( x = 3y^3 + 1 \). This step is crucial in finding the inverse.
3. **Isolating Terms**: We isolate \( y^3 \) by removing constants systematically. We subtract 1 and divide by 3, which transforms \( x = 3y^3 + 1 \) into \( y^3 = \frac{x - 1}{3} \).
4. **Solving for \( y \)**: To solve for \( y \), we take the cube root. This gives: \( y = \sqrt[3]{\frac{x - 1}{3}} \).

• Remember, each algebraic step should maintain the balance of the equation, like keeping scales balanced.
• Correctly following these steps ensures the accurate calculation of the inverse function.

Mastering algebraic manipulations allows you to tackle a variety of mathematical problems with confidence and ease.