Problem 82 - Algebra 2

Question

Find the inverse of each function. Is the inverse a function? \(f(x)=2 x^{3}\)

Step-by-Step Solution

Answer

The inverse function is \(f^-1(x) = (x / 2)^(1/3) \) and it is indeed a function.

Step 1: Swap the Variables

First, replace \(f(x)\) with \(y\) to get the equation \(y = 2x^3\). Swap the \(x\) and \(y\) coordinates so the equation becomes \(x = 2y^3\). This sets up the basis for finding the inverse function.

Step 2: Solve for y

To express the inverse function in terms of \(y\), solve the equation \(x = 2y^3\) for \(y\). Divide both sides by 2 to give: \(x / 2 = y^3\). This shows \(y = (x/2)^(1/3)\).

Step 3: Verify if Inverse is a Function

To verify if the inverse function is a true function, we can perform the horizontal line test. The result of the function is always unique, meaning for any given \(x\), there is only one corresponding \(y\) value. Thus, the inverse is also a function since it passes the horizontal line test.

Key Concepts

Function Verification

Verifying a function, particularly an inverse function, is an essential step to ensure that it truly represents a reversible relationship between variables. When we find an inverse function, what we are actually doing is reflecting the original function across the line \( y = x \). To verify this reflection is indeed correct:

• First, swap the \( x \) and \( y \) in the original equation. This step is crucial as it sets up our new, prospective inverse function.
• Next, solve this new equation for \( y \). The expression you get is a potential inverse function.
• Finally, check if this expression satisfies the original function's requirements. If plugging in the inverse function into the original yields the same input (and vice versa), it confirms correctness.

Verifying helps ensure the outputs match the inputs logically, maintaining the integrity of the function.

Cubic Functions

A cubic function is a polynomial function of degree three, typically written as \( f(x) = ax^3 + bx^2 + cx + d \). They have distinct characteristics:

• The highest power of \( x \) is three, signifying the degree.
• They can have one, two, or three real roots, or solutions, where the curve crosses the \( x \)-axis.
• The graph of a cubic function may have turning points, which could create a local maximum or minimum.

For our case, the function \( f(x) = 2x^3 \) is a simplified cubic function where \( a = 2 \) and other coefficients are zero. It suggests a stretch along the \( y \)-axis. Cubic functions also tend to have a characteristic 'S' shaped curve due to these turns, influencing the range and complexity when finding inverses.

Horizontal Line Test

The horizontal line test is a simple yet effective method for determining whether a function's inverse is still a function. Imagine stretching a horizontal line across the graph of the function:

• If at any y-coordinate the line touches the curve more than once, the inverse does not qualify as a function.
• If each horizontal line intersects the curve at most once, the function's inverse is valid.

An inverse of a function is a function only when it is one-to-one, so it passes the horizontal line test. In our example with \( f(x) = 2x^3 \), the cubic curve means that for each \( y \), there is only one corresponding \( x \). As a result, the horizontal line test confirms that \( f(x) = 2x^3 \) has a function as its inverse. This property makes it possible to find meaningful inverse-related calculations and guarantees predictability across its domain.