Problem 60 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the piecewise-defined function by hand. $$g(x)=\left\\{\begin{array}{ll} x+5, & x \leq-3 \\ 5, & -3 < x <1 \\ 5 x-4, & x \geq 1 \end{array}\right.$$

Step-by-Step Solution

Answer

After completing the steps, we have three connected segments of lines forming the graph of the piecewise-defined function. For x<=-3, the graph resembles 'y=x+5'. For -3=1, it is 'y=5x-4'.

Step 1: Plot Case 'x+5' for 'x<=-3'

This is a straight line equation with slope 1 and y-intercept 5. However, it only applies for 'x<=-3'. So we plot this line only for values of x less than or equal to -3.

Step 2: Plot Case '5' for '-3<x<1'

This represents a horizontal line at y=5, but it only applies between -3-3) and ends with another open circle (indicating x<1).

Step 3: Plot Case '5x-4' for 'x>=1'

This is again a straight line equation but with a slope 5 and y-intercept -4. This line applies for 'x>=1'. So we draw this line for all values of x greater than or equal to 1. Plot the point (1,1) a filled circle (indicating x>=1). Then, for x>1 draw the line with slope 5.

Step 4: Combine all parts

After plotting all three parts, make sure the plot forms a continuous graph faithfully representing g(x).

Key Concepts

Graphing Functions

Graphing functions, especially piecewise functions, is a crucial skill in understanding mathematical relationships visually. In a piecewise function like the one given, different rules apply to different sections of the graph. Each segment can have its own linear equation, making the graph multifaceted and interesting. When graphing functions, follow these steps:

• Clearly identify each piece of the piecewise function and the range of values it covers.
• Use the correct linear equation to compute points within the specified range. For example, for a function part with a given range like \(x \leq -3\), calculate values for multiple \(x\) within that range to accurately plot the segment.
• Graph each segment separately, ensuring you adhere strictly to the inequality constraints, like \(x > -3\) or \(x \geq 1\).
• Be mindful of the endpoints. Sometimes you use open circles to denote the endpoint is not included, whereas filled circles mean the endpoint is part of the solution.

Graphing will be easier if you set up a table with \(x\) values and their corresponding \(g(x)\) values for each section. This practice aids in visualizing the connection and how each piece fits into the puzzle of the overall graph.

Discontinuity

Understanding discontinuity in piecewise functions is an essential concept for sketching accurate graphs. Discontinuity happens when there is a break or a jump in the graph. Here’s how to identify and illustrate discontinuities:

• Assess the endpoints of each section. Particularly look for jumps between one piece and the adjoining segment.
• In our example, the first segment ends at \(x = -3\) and the next starts immediately after \(x > -3\). This implies a discontinuity at \(x = -3\), as represented by an open circle.
• Similarly, the transition between \(-3 < x < 1\) and \(x \geq 1\) may also result in a gap, examined at \(x = 1\) by marking this transition point accurately on the graph.

Visual representation of discontinuity is key. An open circle or a gap indicates values excluded from the graph at specific points. To describe continuity, ensure the transition points smoothly connect or correctly show the interruption.

Linear Equations

Linear equations form the building blocks of the piecewise function. A linear equation involves two variables and graphs as a straight line. The basic form of a linear equation is \(y = mx + c\), where \(m\) is the slope, and \(c\) is the y-intercept.In piecewise functions, each segment can be a different linear equation:

• For the piece \(x + 5\), the slope is \(1\), meaning the line rises one unit up for each unit it moves to the right. Here, it applies for all \(x \leq -3\), so you draw it only till this point.
• The constant segment \(y = 5\) is a unique type of linear equation with a slope of \(0\). This creates a horizontal line, constant and unchanging within its interval.
• The line \(5x - 4\) has a steeper slope of \(5\). A higher slope means steeper ascent; the line rises five units vertically for each unit it moves to the right.

Understanding slopes and intercepts of each linear equation helps in accurately plotting and interpreting piecewise functions. When multiple linear equations coexist in a piecewise function, each line will have its distinct position and slope in a graph.