Problem 34 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=-x^{3}+2 $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = (2 - x)^{1/3} \).

Step 1: Write Equation with y

Start by writing the function with \( y \) as a function of \( x \). So, \( f(x) = y = -x^3 + 2 \).

Step 2: Swap Variables

To find the inverse, swap \( x \) and \( y \). This changes the equation to \( x = -y^3 + 2 \).

Step 3: Solve for y

Rearrange the swapped equation to solve for \( y \):\[ x - 2 = -y^3 \]\[ -x + 2 = y^3 \]\[ y = (-x + 2)^{1/3} \]

Step 4: Write the Inverse Function

Now that \( y \) is isolated, express the inverse function as \( f^{-1}(x) = (2 - x)^{1/3} \).

Key Concepts

Function Notation

Function notation is a way to express mathematical functions in a clear and consistent manner. This notation typically uses a letter, like "\(f\)", followed by a list of variables inside parentheses. For example, \(f(x)\) represents a function named \(f\) with \(x\) as its input. This notation helps distinguish inputs and outputs clearly.

Function notation is beneficial because it eliminates ambiguity. By using this structure, you can effortlessly define a relationship between variables. For instance, in the given exercise, \(f(x)\) illustrates that \(x\) is inputted into the function \(f\) to produce an output. When finding an inverse function, it's crucial to understand function notation to make accurate swaps between variables.

• Standard format: \(f(x) = expression\)
• Inverse function notation: \(f^{-1}(x)\)
• Highlights the dependent and independent variables clearly

Understanding function notation also is essential when dealing with complex expressions, as it simplifies tracking and transforming equations during calculations.

Cubic Function

A cubic function is a polynomial of degree three, typically taking the form \(ax^3 + bx^2 + cx + d\). In this structure, at least one of the coefficients must be non-zero, with \(a eq 0\). In our exercise, the cubic function is \(f(x) = -x^3 + 2\).

Characteristics of a cubic function include having one to three real roots and possibly two turning points. The graph of a cubic function is a curve that can change direction at least once. In some instances, like our exercise, the curve can be a smooth continuous arc without distinct turning points.

• Formula: \(ax^3 + bx^2 + cx + d\)
• Shape: Curve with one to three real roots
• Turns: Up to two turning points depending on coefficients

When finding an inverse of a cubic function, it's crucial to manipulate the equation carefully to solve for \(y\) after swapping variables, as illustrated in the step-by-step solution. This ensures accurate determination of the inverse relationship.

Inverse Operations

Inverse operations are mathematical actions that reverse the effects of each other. Common examples include addition and subtraction, or multiplication and division. When dealing with functions, finding an inverse involves switching inputs and outputs to reverse the effect of the original function.

In our exercise, the original function is \(f(x) = -x^3 + 2\). To find the inverse \(f^{-1}(x)\), you swap \(x\) and \(y\) and solve the equation to express \(y\) as a function of \(x\). This process involves:

• Swapping the roles of \(x\) and \(y\)
• Reorganizing the formula to isolate \(y\)
• Executing inverse operations to solve for the new expression

For instance, starting from \(x = -y^3 + 2\), you reverse operations step-by-step: subtracting 2, then taking the negative, and finally the cube root. Each step is an inverse action that leads to \(y = (2 - x)^{1/3}\). Mastery of inverse operations is pivotal in accurately deriving inverse functions, ensuring clarity and correctness in calculations.