Problem 9 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The machine diagram transforms input \(x\) by \(x-1\) and then the square root operation, yielding \(\sqrt{x-1}\) for \(x \geq 1\).

Step 1: Identify the Function Type

The function given is \( f(x) = \sqrt{x-1} \). This is a square root function with a horizontal shift. The expression inside the square root, \(x-1\), indicates that the function is a transformation of the basic square root function \(\sqrt{x}\).

Step 2: Determine the Domain

For the function \( f(x) = \sqrt{x-1} \), the expression inside the square root must be non-negative to ensure real number outputs. Therefore, we set \( x - 1 \geq 0 \), which simplifies to \( x \geq 1 \). The domain of this function is \([1, \infty)\).

Step 3: Set Up the Machine Diagram Components

In the machine diagram, input \(x\) enters and is first transformed by \(x - 1\), representing the translation of the function. This step is crucial as it shifts the function to the right by 1 unit.

Step 4: Apply the Square Root Transformation

After subtracting 1 from \( x \), apply the square root operation: take the square root of the result to obtain the output \( f(x) \). This represents the vertical stretching of the function.

Step 5: Draw the Diagram

Create the diagram: start with an input \(x\), passing through a box labeled \("Subtract 1"\), leading to a second box labeled \("Square Root"\). This results in the final output \(\sqrt{x-1}\). Ensure to label each part of the process clearly in the diagram.

Key Concepts

Square Root Function

A square root function is a function that involves the square root of a variable. It is typically represented as \( f(x) = \sqrt{x} \). This basic function forms the foundation for understanding more complex qualities and transformations. The graph of \( \sqrt{x} \) starts at the origin \((0,0)\) and continues upward to the right, forming a curve that represents increasing values of \( f(x) \) as \( x \) increases.

One key feature of the square root function is that it only operates on non-negative numbers. This is because taking the square root of a negative number would result in an imaginary number rather than a real number. Therefore, the domain of the basic square root function \( \sqrt{x} \) is \([0, \infty)\). It is essential to remember this when considering more complex functions involving square roots.

Function Transformation

Function transformation refers to the changes made to the graph of a function, altering its position, shape, or orientation on a coordinate plane. There are several types of common transformations that can affect a function:

• Translations: These move the entire graph horizontally or vertically. For example, in the function \( f(x) = \sqrt{x-1} \), the graph of \( \sqrt{x} \) is shifted 1 unit to the right.
• Reflections: Transformations that flip the graph over a line, such as the x-axis or y-axis.
• Stretches and Compressions: These transformations affect the steepness of the graph, either stretching it to make it steeper or compressing it to make it flatter.

Understanding how these transformations work can help you visualize and graph transformed functions quickly. In the context of our function \( f(x) = \sqrt{x-1} \), the horizontal shift is the primary transformation affecting its graph.

Domain of a Function

The domain of a function is the set of all possible input values (or \( x \)-values) that the function can accept without causing any issues. To determine the domain, we think about where the function's expression is defined and real.

For the function \( f(x) = \sqrt{x-1} \), the expression inside the square root, \( x-1 \), must not be negative. This requirement ensures that we do not end up with complex numbers. Solving for \( x \) in the inequality \( x-1 \geq 0 \) gives \( x \geq 1 \). Therefore, the domain of \( f(x) = \sqrt{x-1} \) is \([1, \infty)\). This means that for the function to deliver real, valid results, \( x \) must be at least 1 or greater. Understanding the domain is crucial in graphing functions correctly and predicting their real-world applicability.