Problem 33 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=2 x^{3}-5 $$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \sqrt[3]{\frac{x + 5}{2}}\).

Step 1: Set up the Equation

To find the inverse function, begin by replacing \(f(x)\) with \(y\). Thus, the original equation becomes \(y = 2x^3 - 5\).

Step 2: Solve for x

Express \(x\) in terms of \(y\). Start by isolating \(x^3\): \[ y + 5 = 2x^3 \] Divide both sides by 2: \[ \frac{y + 5}{2} = x^3 \] Finally, take the cube root of both sides to solve for \(x\): \[ x = \sqrt[3]{\frac{y + 5}{2}} \]

Step 3: Swap x and y to get the Inverse Function

Replace \(y\) with \(x\) to denote the inverse function: \[ f^{-1}(x) = \sqrt[3]{\frac{x + 5}{2}} \] This is the inverse of the original function.

Key Concepts

Cube Root

The cube root is a mathematical operation that is used to find a number which, when multiplied by itself twice, gives the original number. For example, the cube root of 8 is 2, because when 2 is multiplied by itself twice (2 x 2 x 2), it equals 8. The notation for the cube root is typically written as \(\sqrt[3]{}\). This symbol can be read as "the cube root of".Understanding cube roots is crucial when working with polynomials and solving equations. In this exercise, the solution involves taking the cube root to isolate the variable \(x\) when finding the inverse function:

• First, rearrange the equation to get \(x^3\) on one side.
• Then, apply the cube root operation to both sides of the equation.

Through these steps, the cube root allows us to solve for \(x\) by "undoing" the cubing operation, which is essential in expressing \(x\) in terms of \(y\) and ultimately finding the inverse function.

Function Notation

Function notation is a way to represent functions in mathematics, making it easier to understand and work with various equations. Usually, functions are denoted with letters like \(f(x)\), \(g(x)\), or \(h(x)\). The notation implies a specific rule where each input \(x\) is associated with exactly one output. In the context of this exercise, function notation helps us define and manipulate the original function and its inverse clearly.In finding the inverse function, we start by replacing \(f(x)\) with \(y\) to make the equation easier to work with. This substitution, \(y = 2x^3 - 5\), does not change the function's rule; it merely simplifies our work as we aim to solve for \(x\). Once we revert back to the inverse form, function notation again simplifies how we express this, using \(f^{-1}(x)\) to denote the inverse function clearly. This highlights the relationship between functions and how changing \(x\) and \(y\) roles can transform a function into its inverse.

Solving Equations

Solving equations is the process of finding the value of variables that make the equation true. It involves a series of steps such as rearranging terms, performing arithmetic operations, and applying the appropriate mathematical functions. In the exercise given, solving for \(x\) is crucial to obtain the inverse function.The original function is \(y = 2x^3 - 5\). Here are the steps involved in solving for \(x\):

• You first isolate \(x^3\) by adding 5 to both sides, resulting in \(y + 5 = 2x^3\).
• Next, divide both sides by 2, simplifying the equation to \(\frac{y+5}{2} = x^3\).
• Finally, apply the cube root to clear the cube on \(x\), leading to a simple expression for x: \(x = \sqrt[3]{\frac{y+5}{2}}\).

Each step of solving ensures that the variable is isolated correctly, which is essential for accurate interpretation and computation of inverse functions. This strategy of systematically breaking down the equation works well for a wide range of mathematical problems.