Problem 4 - Algebra and Trigonometry

Question

Express the rule in function notation. For example, the rule square, then subtract 5 is expressed as the function \(f(x)=x^{2}-5 .\) Take the square root, add \(8,\) then multiply by \(\frac{1}{3}\)

Step-by-Step Solution

Answer

The function is expressed as \(f(x) = \frac{1}{3}(\sqrt{x} + 8)\).

Step 1: Understand the Rule

The rule given describes a sequence of operations that need to be applied to an input. First, you take the square root of the input, then add 8 to the result, and finally multiply the new result by \(\frac{1}{3}\).

Step 2: Apply the Square Root

Start by applying the first operation. If the input is \(x\), then take the square root of \(x\), which is represented as \(\sqrt{x}\).

Step 3: Add 8

Take the result from the previous step \(\sqrt{x}\) and add 8 to it, which can be expressed as \(\sqrt{x} + 8\).

Step 4: Multiply by \(\frac{1}{3}\)

Finally, take the result from Step 3 and multiply it by \(\frac{1}{3}\). This gives us \(\frac{1}{3}(\sqrt{x} + 8)\).

Step 5: Write in Function Notation

Represent the final expression from Step 4 in function notation as \(f(x) = \frac{1}{3}(\sqrt{x} + 8)\).

Key Concepts

Function Composition

Function composition is a way of combining multiple functions into a single function, where the output of one function becomes the input for the next. In simpler terms, it's like following a recipe where each step depends on the results of the previous one. This concept is crucial for understanding how to evaluate and simplify complex rule sets, like the one in our exercise.
Think about it with the following process:

• You start with your input value, say \(x\).
• Apply the first operation, which in this particular case is the square root operation, resulting in \(\sqrt{x}\).
• Take this result and pass it through the next operation, which adds 8, resulting in \(\sqrt{x} + 8\).
• The final operation multiplies this result by \(\frac{1}{3}\), leading to the final expression \(f(x) = \frac{1}{3}(\sqrt{x} + 8)\).

When we say we are composing functions, we are arranging them to be applied successively. It is a powerful way to create new functions with specialized capabilities. Function composition allows for a modular approach, turning complex sequences of mathematical operations into manageable expressions.

Sequential Operations

Sequential operations refer to a structured order of operations performed one after another. Each operation depends on the results from the previous one. When dealing with functions, performing operations in the correct sequence is essential to ensuring an accurate final result.
In our exercise, the sequence is outlined specifically:

• First, apply the square root operation, denoted as \(\sqrt{x}\).
• Next, from the resulting value, add 8 to it, giving \(\sqrt{x} + 8\).
• Lastly, multiply the sum by \(\frac{1}{3}\), forming the function \(\frac{1}{3}(\sqrt{x} + 8)\).

Each step builds upon the previous one, creating a chain effect. If these operations are conducted out of order, the end result will differ, highlighting the importance of performing operations sequentially. This approach is necessary when translating rules into function notation, as it guarantees that the rule is expressed accurately and as intended.

Mathematical Expression

A mathematical expression is a combination of numbers, variables, and operations that collectively represent a value. Understanding how to convert a sequence of operations into a clean mathematical expression is key to grasping function notation.
Our exercise illustrates this by starting with a series of operations and transforming them into a concise expression. Here's how it breaks down:

• The first operation, the square root, is transformed into the mathematical expression \(\sqrt{x}\).
• Adding 8 results in the expression \(\sqrt{x} + 8\).
• Finally, multiplying by \(\frac{1}{3}\) refines it to \(f(x) = \frac{1}{3}(\sqrt{x} + 8)\).

This expression effectively captures the entire rule set in function notation, ensuring clear communication of the operations involved. Mathematical expressions are vital in algebra and higher mathematics as they allow for the concise representation and manipulation of numbers and variables according to defined rules.