Problem 45 - College Algebra

Question

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation. $$ f(x)=\left\\{\begin{array}{l}{|x| \text { if } x<2} \\ {1 \quad \text { if } x \geq 2}\end{array}\right. $$

Step-by-Step Solution

Answer

Graph |x| for \( x<2 \) and y=1 for \( x\geq 2 \); domain is \((-\infty, \infty)\).

Step 1: Identify Sections of the Piecewise Function

The given piecewise function is divided into two sections based on the value of \( x \). The first section is \( f(x) = |x| \) for \( x < 2 \). The second section is \( f(x) = 1 \) for \( x \geq 2 \). Each section needs to be graphed separately and joined according to their conditions.

Step 2: Graph \( f(x) = |x| \) for \( x < 2 \)

The function \( f(x) = |x| \) is a standard absolute value function with a 'V' shape, centered at the origin. For values \( x < 2 \), graph the portion of this function. This part of the graph will increase linearly from \( (0, 0) \) to but not including \( (2, 2) \). Add an open circle at the point \( (2, 2) \) to indicate that this point is not included in this part of the graph.

Step 3: Graph \( f(x) = 1 \) for \( x \geq 2 \)

For \( x \geq 2 \), the function is a constant value \( f(x) = 1 \). Draw a horizontal line at \( y = 1 \) starting from and including the point \( (2, 1) \) extending to the right. Add a closed circle at \( (2, 1) \) to show this point is included.

Step 4: Combine Graph Sections and Label the Domain

Combine the segments from previous steps to create the full graph of the piecewise function. The absolute value portion \( f(x) = |x| \) and the constant portion \( f(x) = 1 \) should connect at \( x = 2 \) with clear open and closed circles. The domain of the function is the set of all real numbers, \((-\infty, \infty)\), as there are defined values for \( f(x) \) for any real number \( x \).

Key Concepts

Graphing Functions

Graphing functions involves plotting a mathematical equation or piecewise definition on a coordinate system to visualize how it behaves across different values of the independent variable, usually denoted as "x". For piecewise functions, which involve multiple expressions for different intervals of the domain, it is essential to graph each piece separately.

Here's a breakdown of graphing a basic piecewise function:

• Each segment of the function must adhere to its specific condition regarding the variable "x". This may involve open or closed circles to indicate whether endpoints are included.
• It's important to clearly distinguish between different intervals and to accurately represent any changes in slope or direction.

In the example given, the piecewise function is defined by an absolute value equation and a constant. Graphing these requires understanding their distinct characteristics and ensuring they are connected properly to reflect their combined domain. This approach ensures an accurate and clear visual representation of the entire function.

Domain Notation

Domain notation expresses all possible input values for which a function is defined. In the case of piecewise functions, each section might have its own domain that, together, form the overall domain of the function.

Here’s how to express domains:

• Use interval notation to represent the continuous range of values. Parentheses, \((, )\), indicate that endpoints are not included, while brackets, \([, ]\), indicate inclusion.
• The domain of the given function is \((-\infty, \infty)\) because there is a defined output for every possible real number "x".

For piecewise functions, it's crucial to check each segment's domain conditions and combine them, if necessary, to describe the domain of the overall function correctly. Pay special attention to transition points, like 2 in the example, where the function’s definition switches.

Absolute Value Functions

Absolute value functions are a specific type of piecewise function characterized by their 'V' shape graph. The absolute value of a number is its distance from zero on the number line, regardless of direction, which means the function is always non-negative.

Understanding absolute value functions involves:

• The basic form is given by \(f(x) = |x|\), which implies \(f(x) = x\) if \(x \geq 0\) and \(f(x) = -x\) if \(x < 0\).
• The graph of \(f(x) = |x|\) is symmetrical around the y-axis, creating the distinct 'V' shape.

In the original piecewise scenario,

• The function \(f(x) = |x|\) is used for \(x < 2\), so only the left side of the 'V' is plotted until it approaches \(x = 2\) without including it.

Understanding this helps in accurately sketching graphs and interpreting shifts or transformations that might affect the function’s visual representation.