Problem 8 - Precalculus: Mathematical for Calculus

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

\( f(x) = \frac{1}{3}(\sqrt{x} + 8) \)

Step 1: Identify Operation Sequence

The rule gives a sequence of operations: 1) take the square root, 2) add 8, and 3) then multiply by \( \frac{1}{3} \). We need to apply these in the given order to a variable \( x \).

Step 2: Apply the Square Root

For any input \( x \), the first step is to take the square root. This is expressed mathematically as \( \sqrt{x} \).

Step 3: Add 8 to the Result

After taking the square root, add 8 to the result. This can be expressed as \( \sqrt{x} + 8 \).

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

Take the result from the previous step and multiply it by \( \frac{1}{3} \). This results in the final expression: \( \frac{1}{3}(\sqrt{x} + 8) \).

Step 5: Write in Function Notation

Express the entire operation as a function \( f \) of \( x \). Thus, the function notation is \( f(x) = \frac{1}{3}(\sqrt{x} + 8) \).

Key Concepts

Operations Sequence

When working with function notation, understanding the order in which operations are performed is crucial. An operations sequence is simply the order in which multiple mathematical operations are applied to a variable or number. In mathematics, following the correct sequence of operations ensures that we get the accurate result every time.

• The initial step is to identify each operation we need to perform. For example, in our exercise, we have three operations: taking the square root, adding 8, and multiplying by \( \frac{1}{3} \).
• Next, it's important to execute these operations in the precise order given in the rule. The sequence is important because changing it might alter the result entirely.

Breaking down the sequence into smaller parts helps you understand and manage more complex equations over time. With practice, handling operations in the correct sequence will become more intuitive.

Mathematical Expression

A mathematical expression represents a combination of numbers, symbols, and operators that denote a specific quantity or action. In the context of function notation, these expressions are rewritten to show the operations applied to a specific variable.
To create a mathematical expression from a rule, you:

• Identify each mathematical operation given by the rule.
• Translate these operations into mathematical symbols.
• Combine these symbols into a structured mathematical expression.

In our exercise, translating the operations sequence gives the expression \( \sqrt{x} + 8 \), and then multiplying the result by \( \frac{1}{3} \) gives the final expression: \( \frac{1}{3}(\sqrt{x} + 8) \). This formulation captures the entire operations sequence in a concise form.

Function of a Variable

The term "function of a variable" refers to a relationship where each input from one set corresponds to exactly one output in another set. Using function notation helps express this relationship in a clear and mathematical way.
Function notation typically uses symbols like \( f(x) \) to denote that \( f \) is a function with the variable \( x \) as input. The output result is what you get after performing all specified operations on \( x \).

• Functions provide a useful way of defining how a variable is transformed through mathematical operations.
• In our example, \( f(x) = \frac{1}{3}(\sqrt{x} + 8) \) shows how any variable \( x \) is transformed.

Function notation is concise yet informative, allowing for complex expressions to be easily communicated and understood.