Problem 59 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the piecewise-defined function by hand. $$f(x)=\left\\{\begin{array}{ll} x+3, & x \leq 0 \\ 3, & 0 < x \leq 2 \\ 2 x-1, & x >2 \end{array}\right.$$

Step-by-Step Solution

Answer

The graph will consist of three parts. The first part is a straight line \( y=x+3 \) for \( x \leq 0 \), ending at the point (0,3). The second part is a horizontal line at \( y=3 \) from \( x=0 \) to \( x=2 \), not including the point at \( x=0 \) but including the point at \( x=2 \). The last part is a straight line \( y=2x-1 \) for \( x>2 \), starting just after the point (2,3).

Step 1: Understand the function pieces and their domains

The given function \( f(x) \) has been broken down into three different smaller functions or pieces that are defined over three different domains or intervals of \( x \).\nThe function \( f(x) \) is defined as:\n \( f(x) = x+3 \), for \( x \) is less than or equal to 0\n \( f(x) = 3 \), for \( x \) is between 0 and 2 (exclusive of 0 and inclusive of 2)\n \( f(x) = 2x-1 \), for \( x \) is greater than 2

Step 2: Plot the first piece of function

The first part of the function is a straight line \( y=x+3 \). It's defined for \( x \) less than or equal to 0. So plot this line for \( x \) less than or equal to 0, with the line ending at the point (0,3), where the value of the function is calculated.

Step 3: Plot the second piece of function

Function \( y=3 \) is a line parallel to the x-axis. It does not depend on the x-values meaning that for all x in its domain, \( y=3 \). As it is defined for \( 0 < x \leq 2 \), plot a horizontal line starting just after the point (0,3) and ending at point (2,3). Note that point (0,3) lies on the first piece, so don't include it.

Step 4: Plot the last piece of function

The last piece of function \( y=2x-1 \) is also a straight line with a gradient 2. As it is defined for \( x>2 \), start plotting this line from a point just after \( x=2 \). At \( x=2 \), the line will intersects y-axis at \( y=2*2-1=3 \), so the line begins just after this point above y=3.

Key Concepts

Function Graphing

Graphing a piecewise function involves breaking it down into smaller parts, where each part covers specific domains. For the given piecewise function, we have three segments. Each segment is a simple function with its own formula and domain. Understanding how to graph these segments helps us visualize how the overall function behaves.

• The first segment, defined by the equation \( y = x + 3 \), is applicable when \( x \leq 0 \). This represents a portion of a line that includes the point where \( x \) equals zero.
• The second segment, \( y = 3 \), remains constant regardless of the \( x \)-value, as long as \( 0 < x \leq 2 \). This appears as a horizontal line on the graph.
• The third segment, defined by \( y = 2x - 1 \), is valid for \( x > 2 \). This part starts just after \( x = 2 \) and continues infinitely to the right.

To plot these, carefully consider where one segment ends and another begins. Each piece transitions smoothly to the next while maintaining its unique behavior or slope.

Linear Equations

Linear equations form the basis for many piecewise functions. A linear equation represents a straight line when plotted on a graph. It can be generally expressed as \( y = mx + c \), where \( m \) is the slope, and \( c \) is the y-intercept.

In the piecewise function provided, two of the segments are linear equations:

• In \( y = x + 3 \), both the slope (\( m = 1 \)) and y-intercept (\( c = 3 \)) influence how the line is formed. This creates an upward slant when graphed for \( x \leq 0 \).
• The equation \( y = 2x - 1 \) shows a sharper ascent since the slope \( m = 2 \). This implies for each unit increase in \( x \), the value of \( y \) increases by 2.

Understanding these components is crucial for correctly sketching or interpreting the graph of a piecewise function. Recognizing the impact of slope and intercept helps discern the nature of each segment. This is especially true when determining how the line progresses within its specific domain.

Domain of a Function

The concept of a function's domain is fundamental to understanding which values a function can accept. Particularly for a piecewise function, each "piece" functions over a specific range of \( x \) values, known as its domain.

• In the piecewise function example, the first piece \( x + 3 \) applies for \( x \leq 0 \), suggesting it includes all values up to and including zero.
• The second piece \( y = 3 \) is defined where \( 0 < x \leq 2 \). Here, the function only equals 3 for values between 0 and 2, with the interval not including 0 but including 2.
• The function \( 2x - 1 \) governs when \( x > 2 \), meaning it covers all values greater than two without an upper limit.

Recognizing each segment's domain highlights the intervals where the function is applicable. It's important to denote exact points, like whether an endpoint is included or excluded. These specify where one part of the function ends and another begins, ensuring accurate graph representation.