Math Problem Statement

What is the piecewise function depicted in the provided graph?

Solution

The graph you provided seems to depict a piecewise function. It is segmented into different sections that appear to have distinct behaviors over specific intervals of the x-axis.

Breakdown of the graph:

  1. For x<0x < 0:
    The graph is a horizontal line along y=1y = -1, meaning the function is constant with f(x)=1f(x) = -1 in this interval. This line segment stops at the point (1,1)(-1, -1) and is closed at that endpoint.

  2. At x=0x = 0:
    There is an open circle at (0,0)(0, 0), meaning the function is not defined at this point (it's a "hole" in the graph).

  3. For 0<x20 < x \leq 2:
    The graph increases linearly between (0,0)(0, 0) (open) and (2,3)(2, 3) (closed), with the function having a positive slope in this region. The slope here suggests that the function is increasing linearly.

Function Interpretation:

This can likely be expressed as a piecewise function:

\begin{cases} -1 & \text{for } x < 0 \\ \text{undefined} & \text{at } x = 0 \\ \text{linear, with slope between } (0, 0) \text{ and } (2, 3) & \text{for } 0 < x \leq 2 \end{cases}$$ Would you like more details on how to formally express the piecewise function, or do you have any other questions? ### 5 Relative Questions: 1. What is the equation for the linear part of the piecewise function between $$x = 0$$ and $$x = 2$$? 2. How would you compute the slope of the linear segment in the second interval? 3. What does the open circle at $$(0, 0)$$ indicate about the function’s value at $$x = 0$$? 4. How can this piecewise function be used to determine continuity or discontinuity? 5. What types of real-life scenarios might be modeled by such a piecewise function? ### Tip: When interpreting graphs with open or closed circles, remember that open circles mean the function is undefined at that specific point. Closed circles indicate the function is defined and takes a value at that point.

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Linear Equations
Graph Interpretation

Formulas

Linear equation formula: y = mx + b
Slope formula: m = (y2 - y1) / (x2 - x1)

Theorems

Piecewise function definition
Continuity and Discontinuity in functions

Suitable Grade Level

Grades 9-12