Math Problem Statement

Write the piecewise function for the graph

Write the piecewise function for the graph.

Solution

The graph provided appears to depict a piecewise function g(x)g(x), composed of three distinct linear segments. Let's analyze each section of the graph:

  1. First segment (leftmost):

    • This is a horizontal line on the interval 9x4-9 \leq x \leq -4 with an open circle at x=9x = -9 and a closed circle at x=4x = -4.
    • The function has a constant value of y=9y = -9 over this interval.
    • Equation: g(x)=9g(x) = -9 for 9<x4-9 < x \leq -4.
  2. Second segment (middle):

    • This segment is an increasing line from (1,4)(1, 4) to (3,8)(3, 8), with closed circles at both endpoints.
    • The slope of this line can be calculated as m=8431=2m = \frac{8 - 4}{3 - 1} = 2, and the equation of the line in point-slope form is: g(x)=2(x1)+4=2x+2g(x) = 2(x - 1) + 4 = 2x + 2
    • Equation: g(x)=2x+2g(x) = 2x + 2 for 1x31 \leq x \leq 3.
  3. Third segment (rightmost):

    • This is another increasing line from (6,2)(6, 2) to (8,6)(8, 6), with an open circle at x=6x = 6 and a closed circle at x=8x = 8.
    • The slope of this line is m=6286=2m = \frac{6 - 2}{8 - 6} = 2, and the equation of the line is: g(x)=2(x6)+2=2x10g(x) = 2(x - 6) + 2 = 2x - 10
    • Equation: g(x)=2x10g(x) = 2x - 10 for 6<x86 < x \leq 8.

The Piecewise Function:

\begin{cases} -9 & \text{for } -9 < x \leq -4 \\ 2x + 2 & \text{for } 1 \leq x \leq 3 \\ 2x - 10 & \text{for } 6 < x \leq 8 \end{cases}$$ Would you like a deeper explanation of any part, or do you have any specific questions? --- Here are 5 related questions: 1. How do you calculate the slope of a line between two points? 2. What is the difference between an open and a closed circle on a graph? 3. How can you derive the equation of a line given two points? 4. What is a piecewise function and how is it used in mathematics? 5. What are the conditions for a function to be continuous? **Tip:** When writing piecewise functions, always check the behavior at the endpoints (open or closed circles) to define the correct inequalities for each segment.

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Linear Equations
Graph Interpretation

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)
Equation of a line: y = mx + b

Theorems

Basic principles of piecewise functions
Slope-intercept form of a line

Suitable Grade Level

Grades 9-11