Math Problem Statement

Graph the following piecewise function on the axes provided: f(x) = { x + 3, for -1 < x ≤ 2 -x + 9, for 2 < x ≤ 6 Click and drag to make a line. Click the line to delete it. Click on an endpoint of a line to change it.

Solution

To graph the given piecewise function, follow these steps:

Function:

  1. f(x)=x+3f(x) = x + 3 for 1<x2-1 < x \leq 2
  2. f(x)=x+9f(x) = -x + 9 for 2<x62 < x \leq 6

Steps to Graph:

1. For f(x)=x+3f(x) = x + 3 (1<x2-1 < x \leq 2):

  • Start by substituting the boundary points:
    • At x=1x = -1: f(1)=1+3=2f(-1) = -1 + 3 = 2
    • At x=2x = 2: f(2)=2+3=5f(2) = 2 + 3 = 5
  • Plot the segment from (1,2)(-1, 2) (open circle) to (2,5)(2, 5) (closed circle).

2. For f(x)=x+9f(x) = -x + 9 (2<x62 < x \leq 6):

  • Substituting the boundary points:
    • At x=2x = 2: f(2)=(2)+9=7f(2) = -(2) + 9 = 7 (open circle at (2,7)(2, 7)).
    • At x=6x = 6: f(6)=(6)+9=3f(6) = -(6) + 9 = 3 (closed circle at (6,3)(6, 3)).
  • Plot the segment from (2,7)(2, 7) (open circle) to (6,3)(6, 3) (closed circle).

Graph:

  1. Draw the first line segment for x+3x + 3, from (1,2)(-1, 2) to (2,5)(2, 5).
  2. Draw the second line segment for x+9-x + 9, from (2,7)(2, 7) to (6,3)(6, 3).

Would you like a detailed illustration or assistance in sketching?

Here are five relevant follow-up questions:

  1. Would you like a table of values for each piece of the function?
  2. Should I provide step-by-step calculations for each segment?
  3. Do you need help identifying open and closed points on the graph?
  4. Are you familiar with how to plot piecewise functions?
  5. Would a digital or graphical representation of this be more helpful?

Tip: Always check the domain boundaries carefully to ensure you plot open and closed circles correctly!

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Linear Functions
Domain and Range
Graphing

Formulas

f(x) = x + 3
f(x) = -x + 9

Theorems

Graphing Linear Equations
Domain and Range Representation

Suitable Grade Level

Grades 8-10