Problem 31 - Algebra A Combined Function

Question

Graph each piecewise-defined function. $$ g(x)=\left\\{\begin{array}{rll} -x & \text { if } & x \leq 1 \\ 2 x+1 & \text { if } & x>1 \end{array}\right. $$

Step-by-Step Solution

Answer

Graph the line \( y = -x \) for \( x \leq 1 \) and \( y = 2x + 1 \) for \( x > 1 \).

Step 1: Understand the Function

The function \( g(x) \) is defined in pieces. For \( x \leq 1 \), the function is \( g(x) = -x \). For \( x > 1 \), the function is \( g(x) = 2x + 1 \). This defines a set of rules for the graph based on the value of \( x \).

Step 2: Graph the First Piece

For the part of the function where \( x \leq 1 \), graph \( g(x) = -x \). This is a straight line with a slope of \(-1\) and passes through the origin (0,0). However, since the condition is \( x \leq 1 \), we end the line at \( x = 1 \). At \( x = 1 \), \( g(1) = -1 \), so we plot a filled dot at (1,-1) to show that this point is included in the graph.

Step 3: Graph the Second Piece

For \( x > 1 \), graph \( g(x) = 2x + 1 \). This is a line with a slope of \(2\) and y-intercept of \(1\). Calculate the value at the boundary, \( x = 1 \), which is \( g(1) = 3 \). Since \( x \) cannot be \( 1 \) in this section, draw an open circle at \( (1, 3) \). Then draw the line for \( x > 1 \), extending towards higher values of \( x \).

Step 4: Combine the Pieces

On a graph, combine both separate pieces, ensuring that the piece for \( x \leq 1 \) is drawn up to and including the point (1, -1), while the piece for \( x > 1 \) begins right after (1, 3) with an open circle, extending to the right. This shows the discontinuity and two distinct functions for each interval.

Key Concepts

Graphing Functions

Graphing functions is a visual way to represent how a function behaves over its domain. When you're working with a piecewise function, you're actually dealing with multiple sub-functions, each defined over a specific interval. To graph these, you need to consider each piece separately.
It's like drawing a patchwork quilt, piece by piece, making sure each piece accurately represents its specific rule. Here's how you can graph a function like

• Identify each piece of the function and its condition. For example, in the function given, we have
  • The line
  • Another line
• Graph each part within its specified range. For the line where
  • Draw it for all
  • Stop the line at
• Combine the pieces on the same graph. Ensure all parts align with their conditions, and you've got a complete picture of

By following these steps, you can clearly and accurately graph piecewise functions, showing how the function behaves differently over its defined intervals.

Function Discontinuity

Not all functions are smooth and continuous. Sometimes, there's a clear jump or gap in the function. This situation is known as function discontinuity. It's essential to understand this concept for graphing piecewise functions, as it indicates where the function does not flow from one piece to another smoothly.
In our piecewise function example, the function has a discontinuity at

• The first piece ends at
  • A solid dot at
• The second piece starts just after
  • An open circle at

This marks a point where the function jumps from one value to another without covering all intermediate values. Discontinuities are important as they highlight transitional points in piecewise functions, and students should mark them clearly when graphing.
By recognizing and marking discontinuities, you gain a deeper understanding of how the function changes at different sections.

Linear Equations

Linear equations form the backbone of many mathematical problems. In the context of piecewise functions, each piece is often defined by its own linear equation. Understanding how to work with these is crucial for graphing and analyzing piecewise functions effectively.A linear equation describes a straight line with the general format \[ y = mx + b \] where \( m \) represents the slope and \( b \) the y-intercept. For example:

• The line \(-x\) has a slope of
  • This graph will slope downwards from left to right, and in our case ends at
• The equation \(2x + 1\) has
  • This line slopes upwards.
    Even though the piece begins right after

To graph each part of a piecewise function, you'll employ your understanding of linear equations. By calculating critical points and slope direction, you can predict and draw the path of each line. Mastering this makes dealing with complex functions more straightforward and intuitive.