Math Problem Statement
Solution
The graph provided shows two functions, which is identified as , and , which appears as a linear piecewise function. We need to determine the formula for by examining the graph:
Observations
- The graph of is made of two distinct linear segments.
- The first segment, from to , rises from to .
- The second segment, from to , rises from to .
Determining the Slopes
-
For the first segment:
- The slope can be calculated as .
- .
- .
- Thus, .
So, the equation for the first segment is (since when , ).
-
For the second segment:
- .
- .
- Thus, .
The equation for the second segment is (since when , ).
Piecewise Function
Thus, the function is a piecewise function that can be defined as:
x + 8 & \text{for } -8 \leq x < 0 \\ 2x & \text{for } 0 \leq x \leq 4 \end{cases}$$ This piecewise function matches the observed segments in the graph. If you need more details or have any questions, please let me know. Here are some related questions you might consider: 1. How would you calculate the x-intercept for each segment of $$g(x)$$? 2. What is the domain and range of the function $$g(x)$$? 3. How would the function $$g(x)$$ change if the line extends beyond $$x = 4$$ and $$x = -8$$? 4. Could $$g(x)$$ represent a real-world scenario? If so, what could it represent? 5. How would you graph the derivative of $$g(x)$$ based on the slopes of each segment? **Tip:** To fully understand the behavior of piecewise functions, it is beneficial to also graph these functions manually to see how they transition between different segments.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Piecewise Functions
Linear Functions
Graph Analysis
Formulas
Slope Formula (m = Δy/Δx)
Piecewise Function Definition
Theorems
Linear Function Properties
Piecewise Function Construction
Suitable Grade Level
Grades 9-12
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