Math Problem Statement

The graphs of f and g are given. Find a formula for the function g.

Solution

The graph provided shows two functions, f(x)f(x) which is identified as x|x|, and g(x)g(x), which appears as a linear piecewise function. We need to determine the formula for g(x)g(x) by examining the graph:

Observations

  1. The graph of g(x)g(x) is made of two distinct linear segments.
  2. The first segment, from x=8x = -8 to x=0x = 0, rises from y=8y = -8 to y=0y = 0.
  3. The second segment, from x=0x = 0 to x=4x = 4, rises from y=0y = 0 to y=8y = 8.

Determining the Slopes

  1. For the first segment:

    • The slope mm can be calculated as m=ΔyΔxm = \frac{\Delta y}{\Delta x}.
    • Δy=0(8)=8\Delta y = 0 - (-8) = 8.
    • Δx=0(8)=8\Delta x = 0 - (-8) = 8.
    • Thus, m=88=1m = \frac{8}{8} = 1.

    So, the equation for the first segment is y=x+8y = x + 8 (since when x=8x = -8, y=0y = 0).

  2. For the second segment:

    • Δy=80=8\Delta y = 8 - 0 = 8.
    • Δx=40=4\Delta x = 4 - 0 = 4.
    • Thus, m=84=2m = \frac{8}{4} = 2.

    The equation for the second segment is y=2xy = 2x (since when x=0x = 0, y=0y = 0).

Piecewise Function

Thus, the function g(x)g(x) 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.

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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