Problem 13 - Algebra and Trigonometry

Question

Draw a machine diagram for the function. $$f(x)=\sqrt{x-1}$$

Step-by-Step Solution

Answer

The machine diagram involves subtraction followed by a square root operation.

Step 1: Understand the Function

The given function is \(f(x) = \sqrt{x-1}\). This function represents the square root of \(x\) minus 1. It is essential to understand that for the square root function to be defined, the expression inside the square root must be greater than or equal to zero.

Step 2: Determine the Domain

To determine where the function is defined, we need \(x-1 \geq 0\). Solving this inequality, we find \(x \geq 1\). This means the function is defined for all \(x\) greater than or equal to 1.

Step 3: Identify the Components of the Diagram

In a machine diagram, think of each step of the function as a part of a machine. The function \(f(x) = \sqrt{x-1}\) has one subtraction operation and one square root operation.

Step 4: Sketch the Machine Diagram

- Start with an input node labeled \(x\).- Connect this to a node that performs the operation \(x - 1\).- The output of this node connects to another node where the square root function is applied. This node should be labeled \(\sqrt{\cdot}\).- The final output node produces \(f(x) = \sqrt{x-1}\).

Step 5: Annotate the Diagram

Ensure that each operation is clearly labeled to show what function is being performed at each step of the machine. This will help anyone reading the diagram understand the process to calculate \(f(x)\) from the input \(x\).

Key Concepts

Function Domain

When we talk about the domain of a function, we're really discussing the set of all possible input values (often represented by "x") that will provide a valid output. To find the domain, we need to determine for which values of 'x' the function is mathematically defined.
For the function \( f(x) = \sqrt{x-1} \), the expression inside the square root, which is \(x-1\), must be non-negative, i.e., \(x-1 \geq 0\).
Solving this inequality gives us \(x \geq 1\). This means the domain of the function \( f(x) = \sqrt{x-1} \) is all real numbers \(x\) such that \(x\) is greater than or equal to 1.
In simpler terms:

• If you plug in any number less than 1 into \(x\), the function won't work because you can't take the square root of a negative number in real numbers domain.
• Therefore, \(x\) must be equal to or greater than 1.

Square Root Function

The square root function, represented as \( \sqrt{x} \), is a function that returns the non-negative square root of a number \(x\). Here's how it works:

• If \(x\) is 25, the square root is 5, because 5 squared (or 5 multiplied by itself) is 25.
• The key requirement for a square root to exist in real numbers is that \(x\) must be zero or positive.

However, our function of interest is \( f(x) = \sqrt{x-1} \):

• Notice the \(-1\) inside the square root. This indicates we first adjust \(x\) by subtracting 1, and then find the square root of the resulting value.

This requirement affects the domain as mentioned in the previous section. Before finding the square root, make sure \(x - 1\) is non-negative for the function to be defined, leading to \(x \geq 1\).

Machine Diagram

A machine diagram helps in visualizing how a function processes inputs to produce outputs. It breaks down each step clearly, so anyone can understand the flow of calculations.
For the function \( f(x) = \sqrt{x-1} \), imagine a machine performing these tasks:

• Input Node: Start by feeding in any number \(x\) from the domain, which is \(x \geq 1\).
• Subtraction Node: The first task the machine performs is to subtract 1 from \(x\), resulting in \(x-1\).
• Square Root Node: Next, the machine processes \(x-1\) through the square root operation, producing the final result \(\sqrt{x-1}\).
• Output: The machine provides the output \( f(x) = \sqrt{x-1} \).

Each step is like a gear in a machine, working in sequence to transform the input to the output.

Function Operations

Function operations involve manipulating functions using addition, subtraction, multiplication, division, and other operations like composition. For our function \( f(x) = \sqrt{x-1} \):

• Subtraction: Begin with subtracting 1 inside the function. This shifts the "starting point" over, impacting both the domain and outcome.
• Square Root: This operation, in simplest terms, means 'what number squared gives this number?'. It takes what result the subtraction yields and transforms it further.

In the context of more complex operations, these steps demonstrate function transformation, showcasing how basic operations create new functions. When combining functions, respecting domain rules is crucial since each operation can influence whether a function is properly defined.