Problem 23 - Calculus with Concepts in Calculus

Question

Sketch the graph of the function. $$ f(x)=\left\\{\begin{array}{l} x \text { for } x<0 \\ 2 x \text { for } x \geq 0 \end{array}\right. $$

Step-by-Step Solution

Answer

The graph consists of two linear segments meeting at the origin: \( f(x) = x \) for \( x < 0 \) and \( f(x) = 2x \) for \( x \geq 0 \).

Step 1: Identify the Piecewise Function Segments

This piecewise function has two segments. The first segment is defined as \( f(x) = x \) for \( x < 0 \) and the second segment is \( f(x) = 2x \) for \( x \geq 0 \). Each segment has distinct linear behavior, so we will consider them separately.

Step 2: Sketch the First Segment

For \( x < 0 \), the function is \( f(x) = x \). This is a linear function with a slope of 1 passing through the origin, but since it is only defined for negative \( x \)-values, it is a line extending from the origin to the left.

Step 3: Sketch the Second Segment

For \( x \geq 0 \), the function is \( f(x) = 2x \). This is another linear function but with a slope of 2. It starts at the origin and extends to the right. This segment is graphically represented by a straight line that is steeper than the first segment.

Step 4: Identify the Point of Intersection

Both segments intersect at the point \( (0, 0) \). This point belongs to the second segment (\( 2x \)) due to the 'greater than or equal to' condition, \( x \geq 0 \).

Step 5: Draw the Overall Graph

Begin by plotting the first segment on the left side of the \( y \)-axis for negative \( x \)-values, showing a line with a slope of 1. Next, draw the second segment starting at the origin, extending to the right with a steeper slope of 2. Ensure the two lines meet at the origin, ensuring the first segment doesn't include this point explicitly.

Key Concepts

Understanding Graph Sketching for Piecewise Functions

Graph sketching involves drawing the visual representation of a function on a coordinate plane. When dealing with piecewise functions, the process requires sketching each segment of the function according to its specific domain.
Piecewise functions are distinct as they are defined by different expressions within different intervals of the independent variable.

• Start by identifying each segment of the function along with its domain.
• In this case, you have two separate segments: one for negative values and one for non-negative values of x.
• Next, graph each segment of the function within its respective domain.
• Check if there are any points of intersection or junction points by evaluating where the segments meet.

For our given function, the point at which the two segments connect (0,0) is crucial, as it defines a transition between the segments.

Introduction to Linear Functions

Linear functions are the simplest type of function, represented by a straight line in graphical form. A linear equation is generally expressed in the form \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
When analyzing piecewise functions, you'll often find linear functions in each interval.
For the given function, the two segments \( f(x) = x \) and \( f(x) = 2x \) are both linear functions:

• \( f(x) = x \) has a slope of 1 and is defined for \( x < 0 \).
• \( f(x) = 2x \) has a slope of 2 for \( x \geq 0 \).

Linear functions are straightforward to graph, as they require plotting just the slope and intercept, resulting in a straight line.

Slope Analysis in Graph Sketching

The slope of a function is a measure of its steepness and direction. In mathematical terms, it's the ratio of the vertical change to the horizontal change between two points on the line, often expressed as \( \frac{rise}{run} \).
Slope analysis plays a critical role in understanding and sketching piecewise functions.

• For \( f(x) = x \), the slope is 1, indicating a moderate incline moving from left to right.
• In contrast, \( f(x) = 2x \) has a slope of 2, showcasing a steeper line.
• The difference in slopes is crucial for sketching: a steeper slope means a greater angle relative to the x-axis.

Always consider the slope when sketching, as it will help determine the angle and direction of each line segment in the piecewise function.