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

Step 1: Identify the Function Components

The problem describes three operations: taking the square root, adding 8, and then multiplying by \( \frac{1}{3} \).

Step 2: Express Taking the Square Root

The first operation is to take the square root of \( x \). In function notation, this can be written as \( \sqrt{x} \).

Step 3: Add 8 to the Result

Next, add 8 to the result of the square root. This modifies the expression to \( \sqrt{x} + 8 \).

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

Finally, multiply the entire expression by \( \frac{1}{3} \). This gives the function: \( f(x) = \frac{1}{3}(\sqrt{x} + 8) \).

Key Concepts

Square Root

The square root operation is a fundamental concept in mathematics that essentially reverses the process of squaring a number. When taking the square root of a number, you are looking for a value that, when multiplied by itself, gives the original number. In simpler terms, the square root of 9 will be 3, because 3 multiplied by itself equals 9. The square root is represented by the radical symbol \( \sqrt{} \).

Here are a few important points to remember about square roots:

• Every positive number has two square roots: one positive and one negative. However, the principal square root (usually referred to just as \( \sqrt{} \)) is the non-negative one.
• The square root of 0 is 0.
• Taking the square root is the opposite operation of squaring.

In our function expression, starting by taking the square root is key to transforming \( x \) into a form that can be manipulated further with arithmetic operations.

Arithmetic Operations

Arithmetic operations are the basic operations you perform on numbers: addition, subtraction, multiplication, and division. These operations are crucial for establishing mathematical expressions and functions, helping you transform numbers in various ways to achieve desired results.

In our function example, two main arithmetic operations are in play:

• Adding 8: Adding a number increases its value by that amount. For example, adding 8 to our square root expression ensures we adjust the outcome further from its initial value.
• Multiplying by \( \frac{1}{3} \): Multiplication by fractions is another form of scaling or adjusting a number. By multiplying the entire expression by \( \frac{1}{3} \), the value is scaled down by a factor of three, effectively spreading out the range of our function.

Understanding how these operations work individually and together helps clarify their impact on the expression and the resulting function. These operations combined transform the input \( x \) through a series of steps that modify its size and position on the number line.

Function Composition

Function composition involves taking two or more functions and combining them to form a new function. It is like performing several operations sequentially in a defined order, where the output of one function becomes the input of the next. Function composition is a foundational concept especially in calculus and algebra, allowing complex transformations to be represented elegantly.

The function in our exercise is composed of several operations on \( x \), namely taking the square root, adding 8, and then multiplying by \( \frac{1}{3} \).

• First operation: \( \sqrt{x} \) – transforms the input \( x \) into its square root.
• Second operation: \( \sqrt{x} + 8 \) – takes the result and increases it by 8.
• Third operation: \( \frac{1}{3}(\sqrt{x} + 8) \) – finally, scales the whole expression by a third.

Each operation relies on the prior, demonstrating how each is essential to the overall function. This composition results in the function \( f(x) = \frac{1}{3}(\sqrt{x} + 8) \). Understanding function composition is beneficial for understanding complex mathematical models and their behavior.