Problem 42 - Algebra and Trigonometry

Question

Sketch the graph of the piecewise defined function. $$ f(x)=\left\\{\begin{array}{ll}{2 x+3} & {\text { if } x<-1} \\ {3-x} & {\text { if } x \geq-1}\end{array}\right. $$

Step-by-Step Solution

Answer

The graph is composed of two lines: \( y = 2x + 3 \) for \( x < -1 \), and \( y = 3 - x \) for \( x \geq -1 \).

Step 1: Identify the Intervals of the Function

The function is defined in two parts: \( 2x + 3 \) for \( x < -1 \) and \( 3 - x \) for \( x \geq -1 \). Each part is associated with a specific interval on the x-axis.

Step 2: Evaluate the Function for Key Points

Determine the key points by calculating the function values at and around \( x = -1 \). For \( x = -1 \), \( f(x) \) is computed from the second part, so \( f(-1) = 3 - (-1) = 4 \). Also, compute a point as \( x \) approaches \( -1 \) from the left for the first function part: \( f(-1^-) = 2(-1) + 3 = 1 \).

Step 3: Sketch the Graph for \( x < -1 \)

Plot the line \( y = 2x + 3 \). Since this part is valid for \( x < -1 \), it should not include \( x = -1 \). This graph is a straight line with a slope of 2 and a y-intercept at \( (0, 3) \).

Step 4: Sketch the Graph for \( x \geq -1 \)

Plot the line \( y = 3 - x \). This line exists for \( x \geq -1 \) and includes the point where \( x = -1 \) with value \( f(-1) = 4 \). The graph of this part is a straight line with a slope of -1 starting from point \( (-1, 4) \).

Step 5: Combine the Graphs of the Two Intervals

Ensure both graph lines are plotted correctly: the first part as a dashed line up to \( x = -1 \) and the second part starting solidly from \( x = -1 \) and being continuous thereafter. Note the discontinuity at \( x = -1 \) since \( f(x) rightarrow f(-1) \) from the left.

Step 6: Verify Discontinuity and Mark Points

Highlight the open endpoint of \( ( -1, 1 ) \) on the first graph part, and the closed endpoint of \( (-1, 4) \) on the second graph part, to indicate the function's discreteness at \( x = -1 \).

Key Concepts

Graphing Piecewise Functions

Piecewise functions are a type of function defined by multiple expressions, each corresponding to a particular interval of the domain. To graph a piecewise function, you follow the rules set by each piece over their respective intervals. This particular process involves:

• Identifying the different pieces of the function and their applicable intervals.
• Graphing each segment separately, taking care of open and closed endpoints.
• Combining these graphs on the same coordinate plane.

For the function given in the exercise, it's defined in two parts:

• The first part, \( y = 2x + 3 \), is valid for \( x < -1 \). This means it doesn't include the point where \( x = -1 \), making it have an open circle at its boundary.
• The second part, \( y = 3 - x \), applies for \( x \geq -1 \). This part includes the boundary, \( x = -1 \), starting solidly at this point.

A good strategy is to use different styles, like dashed or solid lines, to denote portions included in each interval more clearly. This helps highlight the transition points between the intervals and ensures a neat and comprehensive representation of the entire function.

Discontinuity in Functions

Discontinuity in functions refers to points where a function is not continuous. For piecewise functions, discontinuities often occur at the transition points between pieces. This can happen when the limit of a function as it approaches a point from one side does not equal the limit as it approaches from the other, or it might not equal the function's actual value at that point.

In our exercise, the transition point is at \( x = -1 \). For \( x = -1 \), there is a key discontinuity because:

• The value of the function from the first part approaches \( 1 \) as \( x \to -1^- \) (from the left).
• The value of the function from the second part starts at \( 4 \) for \( x = -1 \).

This results in a jump from \( 1 \) to \( 4 \) at \( x = -1 \). Such a discontinuous jump indicates that the function is not continuous at that point, characterized by an open circle (\( -1, 1 \)) that shows the "missing" value, and a closed circle (\( -1, 4 \)) that signifies the defined function value.

Linear Equations

Linear equations describe relationships where each change in one quantity produces a proportional change in another. They graph as straight lines, defined by their slopes and y-intercepts. For our exercise, both pieces of the piecewise function represent linear equations:

• The first piece, \( y = 2x + 3 \), has a slope of 2 and a y-intercept of 3. This means for every unit change in \( x \), \( y \) increases by 2, and the line starts at the point \( (0, 3) \) on the y-axis.
• The second piece, \( y = 3 - x \), has a slope of -1, indicating that \( y \) decreases by 1 for every unit increase in \( x \). Here, the line intercepts the y-axis at point \( (0, 3) \), but extends backwards to start at \( (-1, 4) \).

Understanding these linear equations is key to accurately plotting each segment on a graph. Notice that each segment fits snugly on the coordinate plane, dictated by its linear equation properties, which help in maintaining the function's overall structure.