Problem 3 - Algebra and Trigonometry

Question

A function \(f\) has the following verbal description: "Multiply by \(3,\) add \(5,\) and then take the third power of the result." (a) Write a verbal description for \(f^{-1} .\) (b) Find algebraic formulas that express \(f\) and \(f^{-1}\) in terms of the input \(x .\)

Step-by-Step Solution

Answer

(a) "Take the cube root, subtract 5, divide by 3." (b) \( f(x) = (3x + 5)^3 \), \( f^{-1}(x) = (\sqrt[3]{x} - 5)/3 \).

Step 1: Understand the Function

The function \( f(x) \) is verbally described as: "Multiply the input by 3, add 5, and then take the third power of the result." This means we start with an input \( x \), perform these operations sequentially to get \( f(x) \).

Step 2: Write the Function Algebraically

From the description: First, multiply by 3, resulting in \( 3x \). Then, add 5, resulting in \( 3x + 5 \). Finally, take the cube of the result: \( (3x + 5)^3 \). Thus, the function is \( f(x) = (3x + 5)^3 \).

Step 3: Transition to Find the Inverse

To find the inverse function \( f^{-1} \), we need to reverse each step of \( f(x) \): starting with the cube root, then subtracting 5, and finally dividing by 3.

Step 4: Reverse the Operation: Cube

Start by taking the cube root of \( y = (3x + 5)^3 \). So we have: \( \sqrt[3]{y} = 3x + 5 \).

Step 5: Reverse the Operation: Subtraction

Next, subtract 5 from both sides: \( \sqrt[3]{y} - 5 = 3x \).

Step 6: Reverse the Operation: Division

Finally, divide by 3 to solve for \( x \): \( x = (\sqrt[3]{y} - 5)/3 \). This gives us the inverse function: \( f^{-1}(x) = (\sqrt[3]{x} - 5)/3 \).

Step 7: Write the Verbal Description of the Inverse

The inverse function \( f^{-1} \) can be described verbally as: "Take the cube root of the input, subtract 5, and then divide by 3."

Key Concepts

Function Operations

Function operations involve sequential processes performed on a given value or input. When you have a function like \( f(x) = (3x + 5)^3 \), this signifies performing specific mathematical steps. These operations are in a particular order: first, you multiply the input by 3, then add 5 to the result. Finally, you take the cube of this entire expression. Each function comprises a series of these operations that transform the input \( x \) into an output \( f(x) \).

Understanding function operations is crucial for manipulating or reversing functions, like finding the inverse. The inverse function \( f^{-1}(x) \) achieves the opposite transformations, posing a fascinating aspect of math called reversibility. To find \( f^{-1}(x) \), we worked backwards through the operations of \( f(x) \): taking the cube root first, then subtracting 5, and dividing by 3. This sequence ensures that \( f(f^{-1}(x)) = x \) and vice versa, preserving the essence of function operations.

Algebraic Expressions

Algebraic expressions are like recipes for performing calculations. They describe specific mathematical steps to perform on a variable. In the function \( f(x) = (3x + 5)^3 \), we interpret and rewrite the operations described verbally into a structured algebraic expression. This expression becomes a powerful tool for quickly calculating the function's output for any given input \( x \). The structure of this expression provides clarity and allows us to see the concise nature of mathematical operations.

To reverse this expression and find \( f^{-1}(x) \), we must unravel the operations step-by-step to express \( x \) in terms of \( f^{-1}(x) \). It's a systematic undoing of the initial steps where we start with \( f(x) \). Algebraically, this involves taking a cube root, then manipulating it with subtraction and division expressions. The inverse function \( f^{-1}(x) = (\sqrt[3]{x} - 5)/3 \) again highlights how useful algebraic expressions are. They succinctly capture the series of mathematical operations needed to relate inputs and outputs.

Verbal Description of Functions

A verbal description of functions provides an easy way to understand the process a function follows. Instead of diving into algebra from the start, verbal descriptions tell the story of the function's behavior. For example, saying "multiply by 3, add 5, and take the cube" outlines the operations involved in \( f(x) \) in a sequence.

Additionally, describing inverse functions verbally, such as "take the cube root, subtract 5, and divide by 3," helps visualize the backward process of achieving the original input from a given output. Transforming verbal descriptions into concrete algebraic form can be challenging, but it ensures you fully grasp the function's inner workings.

Verbal descriptions act as intuitive guides that can demystify functions, showing each step individually to help any learner comprehend complex mathematical transformations. They serve as a foundation for understanding both the direction and the inverse processes involved in a function.