Problem 51 - Intermediate Algebra

Question

Each of the following functions is one-to-one. Find the inverse of each function and express it using \(f^{-1}(x)\) notation. $$ f(x)=2 x^{3}-3 $$

Step-by-Step Solution

Answer

\( f^{-1}(x) = \sqrt[3]{\frac{x + 3}{2}} \)

Step 1: Understand the problem

We are given a function \( f(x) = 2x^3 - 3 \) and need to find its inverse, expressed as \( f^{-1}(x) \). The function is one-to-one, so an inverse exists.

Step 2: Set up the equation

To find the inverse, we start by setting \( y = f(x) \). Therefore, \( y = 2x^3 - 3 \). Our goal is to solve for \( x \) in terms of \( y \).

Step 3: Isolate the cubic term

First, add 3 to both sides to start isolating the cubic term. We get \( y + 3 = 2x^3 \).

Step 4: Solve for the cubic term

Next, divide both sides by 2 to further isolate \( x^3 \): \( x^3 = \frac{y + 3}{2} \).

Step 5: Solve for x

Take the cube root of both sides to solve for \( x \): \( x = \sqrt[3]{\frac{y + 3}{2}} \).

Step 6: Write the inverse

Now that we have \( x \) in terms of \( y \), the function \( f^{-1}(x) \) is \( f^{-1}(x) = \sqrt[3]{\frac{x + 3}{2}} \).

Key Concepts

One-to-One Functions

One-to-one functions are special types of functions where each input corresponds to a unique output and vice versa. In simpler terms, no two different inputs will ever produce the same output. This property is crucial for a function to have an inverse, which is what we're exploring here.
For instance, consider the function given in the problem: \( f(x) = 2x^3 - 3 \). It is a one-to-one function because if you take any two different \( x \) values, their outputs will be different. This is typically verified by checking that the function is either always increasing or always decreasing over its entire domain.

• An important test for one-to-one functions is the horizontal line test. If no horizontal line intersects the graph of the function more than once, the function is one-to-one.
• In our case, the cubic equation causes the function \( f(x) = 2x^3 - 3 \) to be always increasing, which confirms its one-to-one nature.

Consequently, this property allows us to find an inverse function, as each output uniquely traces back to an input.

Cubic Equations

Cubic equations are polynomial equations of degree three. The general form of a cubic equation is \( ax^3 + bx^2 + cx + d = 0 \). In the exercise, our function \( f(x) = 2x^3 - 3 \) is such a cubic equation.
Cubic equations can have varying shapes, but they typically pass through three major phases: quick rise, turning point, and then settling into another trajectory, depending on the coefficients. For simpler cubic functions, like the one given, no turning points disrupt the function's increasing or decreasing behavior.
In the solution, the key is to manipulate the equation to make it possible to solve for \( x \) in terms of \( y \). We start by isolating the cubic term \( x^3 \):

• Add 3 to both sides of \( y = 2x^3 - 3 \), resulting in \( y + 3 = 2x^3 \).
• Divide by 2 to further isolate the cubic term, giving \( x^3 = \frac{y + 3}{2} \).

This manipulative process highlights the flexibility and algebraic characteristics of cubic equations. By being able to express \( x^3 \) in terms of \( y \), we make the calculation of inverses easier.

Function Notation

Function notation is a straightforward but powerful way of expressing relationships between variables. The notation \( f(x) \) represents a function named "f" evaluated at "x". This notation improves clarity and allows for easy manipulation and calculation.
In reverse, finding an inverse function involves not just reversing operations but also expressing it clearly in function notation. Our goal in the problem was to express the inverse of \( f(x) = 2x^3 - 3 \) in the form of \( f^{-1}(x) \).
After manipulating the original function and solving for \( x \) in terms of \( y \), we replace \( y \) back with \( x \) in the final expression for the inverse.
Finally, the inverse is neatly written in function notation as \( f^{-1}(x) = \sqrt[3]{\frac{x + 3}{2}} \). This standardization is crucial in mathematics:

• Helps communicate the function's behavior clearly.
• Indicates the specific input-output relationship being reversed, signifying that \( f^{-1}(x) \) produces the original input value, "undoing" the effect of \( f(x) \).

Function notation, therefore, acts as a universal language in mathematics, aiding in consistent understanding and expression of inverse relationships.