Question

(a) Write a function of \(x\) that performs the operations described. (b) Find the inverse and describe in words the sequence of operations in the inverse. Subtract 5 , divide by 2 , and take the cube root.

Short answer

#Answer# (a) \(f(x) = \sqrt[3]{\frac{x-5}{2}}\) (b) Inverse function: \(f^{-1}(x) = 2x^3 + 5\); operations in the inverse are: cube, multiply by 2, and add 5.

Step-by-step solution

(Part (a) Define the function)

To define a function that represents the given operations, we will apply each operation successively on x. The operations are: 1. Subtract 5: \(x - 5\) 2. Divide by 2: \(\frac{x-5}{2}\) 3. Take the cube root: \(\sqrt[3]{\frac{x-5}{2}}\) So, the function will be: \(f(x) = \sqrt[3]{\frac{x-5}{2}}\)

(Part (b) Inverse function)

To find the inverse function, we will replace \(f(x)\) with \(y\) and swap \(x\) and \(y\): \(y = \sqrt[3]{\frac{x-5}{2}}\) Now, we will solve for \(x\): Step 1: Cube both sides: \(y^3 = \frac{x-5}{2}\) Step 2: Multiply both sides by 2: \(2y^3 = x-5\) Step 3: Add 5 to both sides: \(2y^3 + 5 = x\) So, the inverse function will be: \(f^{-1}(x) = 2x^3 + 5\)

(Part (b) Describe inverse operations)

The sequence of operations in the function's inverse are: 1. Cube x: \(x^3\) 2. Multiply by 2: \(2x^3\) 3. Add 5: \(2x^3 + 5\) Thus, the inverse function performs the opposite operations in reverse order: cube, multiply by 2, and add 5.

Key concepts

Cube Roots

Cube roots are an essential aspect of this exercise as they are key in forming the function from the original problem. A cube root is an operation that "undoes" cubing. When you take the cube root of a number, you are determining which number, when multiplied by itself twice, results in the original number. For example, the cube root of 8 is 2, because when 2 is multiplied by itself twice \(2 \times 2 \times 2 = 8\).
In the context of a function like \(f(x) = \sqrt[3]{\frac{x-5}{2}}\), the cube root is applied as the final step to complete the series of operations. It is represented by the radical sign with a small three, or alternatively written using an exponent of one-third: \(x^{1/3}\).
Understanding cube roots is crucial for working both with and finding the inverse of functions that include such operations, providing a powerful tool to solve equations and manipulate expressions.

Mathematical Operations

Mathematical operations form the building blocks for creating functions and finding their inverses. The operations in the problem can be understood step-by-step:

• Subtraction: The first operation is subtracting 5 from \(x\). This can be viewed as shifting the value to the left on a number line, which moves you lower than \(x\).
• Division: After subtraction, the result is divided by 2. Division reduces the magnitude of a number, effectively compressing the result. The operation \(\frac{x-5}{2}\) decreases the result of the subtraction operation by half.
• Cube Root: Finally, take the cube root, which transforms the result into the function \(f(x)\). This operation is crucial for creating non-linear patterns in mathematical modeling.

These operations, when applied step-by-step, allow us to manipulate \(x\) into the desired output. Understanding how they work individually and in sequence is vital in creating and manipulating mathematical models.

Function Composition

Function composition refers to the process of applying one function to the results of another. In this exercise, we compose three operations into a single function \(f(x)\).
Each operation contributes to a part of the composition:

• The subtraction function \(g(x) = x-5\) is applied first, altering the variable \(x\).
• The result of this is fed into the division function \(h(x) = \frac{x}{2}\), scaling it down.
• Finally, the cube root function \(r(x) = \sqrt[3]{x}\) is applied, transforming the combined result of the previous operations.

Function composition is a powerful concept that allows us to simplify and solve complex real-world problems by breaking them down into smaller and more manageable tasks. It's like putting together a puzzle, where each piece is essential for forming the whole picture. Understanding how to compose and decompose functions is fundamental in mastering many areas of mathematics.