which piecewise function is shown in the graph

3 min read 11-03-2025

which piecewise function is shown in the graph

Understanding piecewise functions and how to represent them graphically is crucial in mathematics. This article will guide you through the process of identifying the correct piecewise function represented by a given graph. We'll cover key concepts, step-by-step analysis, and examples to solidify your understanding.

Understanding Piecewise Functions

A piecewise function is a function defined by multiple sub-functions, each applicable over a specified interval of the domain. These intervals are distinct and don't overlap. The graph of a piecewise function will therefore consist of separate sections, each corresponding to one of its sub-functions. Each section will have its own equation.

Identifying the correct piecewise function from a graph involves determining:

The equations of the individual sub-functions: What are the formulas for each line segment or curve?
The intervals of the domain: For what x-values does each sub-function apply?

Step-by-Step Analysis: Deciphering the Graph

Let's outline a systematic approach to analyze a graph and determine its corresponding piecewise function:

Step 1: Identify the Sub-functions

Carefully examine the graph. How many distinct sections are there? Each section represents a different sub-function. Note if the sections are straight lines (linear functions) or curves (quadratic, cubic, etc.).

Step 2: Determine the Equation for Each Sub-function

For each section:

Linear Functions: Find two points on the line. Use the slope-intercept form (y = mx + b) or point-slope form (y - y1 = m(x - x1)) to determine the equation. Remember m represents the slope and b the y-intercept.
Other Functions: For curves, the process is more involved. You may need more points to determine the type of function (quadratic, cubic, etc.) and use techniques like systems of equations or regression analysis.

Step 3: Define the Intervals

For each sub-function, specify the interval of the x-values where that sub-function is applicable. Note the endpoints of each section on the x-axis. Use inequalities (e.g., x < 2, 2 ≤ x ≤ 5, x > 5) to define these intervals. Pay close attention to whether the endpoints are included (closed circles) or excluded (open circles).

Step 4: Write the Piecewise Function

Finally, combine the equations and intervals to express the piecewise function in the standard notation:

f(x) = {
  equation1,  if condition1
  equation2,  if condition2
  equation3,  if condition3
  ...
}

Remember to use the correct inequalities to indicate which equation applies to which interval.

Example: Identifying a Piecewise Function

Let's say a graph shows:

A horizontal line segment at y = 2 for x values between -3 and 0 (inclusive).
A line segment from (0,2) to (2,4).
A vertical line segment at x = 2 from y = 4 to y = 1.
Another line segment at (2,1) to (4,3)

Solution:

Sub-functions: We have four distinct sections. The first and third are horizontal and vertical lines respectively. The other two are diagonal lines.
Equations:
- Horizontal Line: y = 2
- Diagonal Line 1: (Using points (0,2) and (2,4)) slope = 1, equation: y = x + 2
- Vertical Line: x = 2
- Diagonal Line 2: (Using points (2,1) and (4,3)) slope = 1, equation: y = x -1
Intervals:
- y = 2: -3 ≤ x ≤ 0
- y = x + 2: 0 < x < 2
- x = 2: 1 ≤ y ≤ 4
- y = x - 1: 2 < x ≤ 4
Piecewise Function:

f(x) = {
  2,           if -3 ≤ x ≤ 0
  x + 2,       if 0 < x < 2
  x-1, if 2 < x <=4
}

Note that the vertical line at x=2 is not a function in itself, as it fails the vertical line test. Therefore, we only include the equations for the other line segments in the piecewise function definition.

Conclusion

Identifying the piecewise function represented by a graph requires careful observation and a systematic approach. By following the steps outlined above, you can accurately determine the equations and intervals, ultimately defining the correct piecewise function. Practice with different graphs to strengthen your understanding of this important concept in mathematics. Remember to always check your work! Does your piecewise function accurately reflect all sections of the graph?