Problem 28 - Algebra A Combined Function

Question

Graph each piecewise-defined function. $$ f(x)=\left\\{\begin{aligned} 3 x & \text { if } \quad x<0 \\ x+2 & \text { if } \quad x \geq 0 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The function is graphed with a dashed line for \( x < 0 \) as \( f(x) = 3x \), and a solid line for \( x \geq 0 \) as \( f(x) = x + 2 \).

Step 1: Understand the Function

The function is piecewise-defined, meaning it is split into two parts. For each different piece, a different formula is used: the first part is defined as \( f(x) = 3x \) for \( x < 0 \) and the second part is defined as \( f(x) = x + 2 \) for \( x \geq 0 \).

Step 2: Graph the First Piece

For \( x < 0 \), we use \( f(x) = 3x \). This is a linear function with a slope of 3. Use a dashed line to graph this part for values to the left of the y-axis (\( x = 0 \)) since the point at \( x = 0 \) is not included in this piece.

Step 3: Plot a Few Points for the First Piece

Select values of \( x \) less than 0, such as \( x = -1 \) and \( x = -2 \). Calculate corresponding \( f(x) \) values: \( f(-1) = 3(-1) = -3 \) and \( f(-2) = 3(-2) = -6 \). Plot these points: \((-1, -3)\) and \((-2, -6)\). Extend this with a dashed line toward the left.

Step 4: Graph the Second Piece

For \( x \geq 0 \), use \( f(x) = x + 2 \). This is also a linear function but with a slope of 1 and y-intercept of 2. Use a solid line starting from \( x = 0 \), including the point \( x = 0 \).

Step 5: Plot a Few Points for the Second Piece

Choose values of \( x \) starting from 0, such as \( x = 0 \), \( x = 1 \), and \( x = 2 \). Calculate \( f(0) = 0 + 2 = 2 \), \( f(1) = 1 + 2 = 3 \), and \( f(2) = 2 + 2 = 4 \). Plot these points: \((0, 2)\), \((1, 3)\), and \((2, 4)\), and draw a solid line through these points extending to the right.

Key Concepts

Graphing Linear Functions

Graphing linear functions is essential in understanding how to visually represent relationships between variables. A linear function is typically in the form of \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For these functions, each value of \(x\) yields exactly one value of \(y\), forming a straight line on a Cartesian plane. To graph a linear function:

• Identify the slope \(m\), which indicates the steepness of the line. A positive slope means the line rises as \(x\) increases, while a negative slope means it falls.
• Determine the y-intercept \(b\), which is the point on the y-axis where the line crosses. Graph this point first.
• Use the slope to find another point. Starting from the y-intercept, move according to the rise (numerator) over run (denominator). For example, if \(m = 3\), move 3 units up and 1 unit right.
• Draw the line through these points, extending it across the plane.

Graphing becomes slightly more complex with piecewise functions, where multiple linear equations define different sections of the graph. For example, the function \( f(x) = 3x \) for \( x < 0 \) is graphed using a dashed line because the point at \( x = 0 \) does not belong to this interval.

Functions and Inequalities

In mathematics, functions and inequalities often work hand-in-hand to define relationships where certain conditions constrain how values interact. An inequality is a statement that describes the relative size or order of two values, using symbols such as \(<\), \(>\), \(\leq\), and \(\geq\). In piecewise functions, these inequalities are essential:

• The symbol \(<\) or \(\leq\) identifies the conditions under which a specific part of the function applies.
• In our example, for the piece \( f(x) = 3x \), \( x < 0 \) means the line only includes points where \( x \) is less than zero.
• Conversely, \( x \geq 0 \) describes the range for \( f(x) = x + 2 \), encompassing zero and all positive values of \( x \).

In any graph, the transition point—here at \( x = 0 \)—divides the inequalities, marking where one condition gives way to another. Recognizing this helps us understand which section of the formula applies and ensures accuracy when plotting each segment of the function.

Mathematical Plotting

Mathematical plotting is the process of representing data or functions on a graph. This visualization is crucial for analyzing and interpreting mathematical relationships. When plotting piecewise functions, follow a clear set of steps:

• First, determine all segments of the function, noting the conditions for each.
• Graph each segment individually, ensuring boundaries are represented by dashed or solid lines depending on whether the endpoint is included.
• Choose specific points within the domain of each piece to plot accurately on the graph, like \( (-1, -3) \) and \( (0, 2) \) from the example.
• Connect the plotted points within each section smoothly, ensuring that transitions between different parts are clearly marked.

Beyond merely mechanical steps, effective plotting improves comprehension by making abstract concepts more concrete and identifying patterns or changes in behavior across different regions of the graph.