Math Problem Statement

Create a piecewise function from the graph based on the given intervals and function behavior.

Solution

To create a piecewise function from the graph, we need to determine the function's behavior across different intervals of the x-axis (age in years). Here's how we can write the piecewise function:

  • From age 0 to 7, f(x)=2f(x) = 2
  • From age 7 to 10, f(x)=−6x+44f(x) = -6x + 44 (for x between 7 and 10, the slope is calculated as (1−2)/3(1 - 2)/3)
  • From age 10 to 13, f(x)=4x−35f(x) = 4x - 35 (for x between 10 and 13, the slope is calculated as (3−1)/3(3 - 1)/3)
  • From age 13 to 17, f(x)=3f(x) = 3

The piecewise function is:

\begin{cases} 2, & 0 \leq x < 7 \\ -6x + 44, & 7 \leq x < 10 \\ 4x - 35, & 10 \leq x < 13 \\ 3, & 13 \leq x \leq 17 \end{cases}$$

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Linear Functions
Slope Calculation

Formulas

Piecewise function definition
Linear function formula: f(x) = mx + b

Theorems

Slope formula

Suitable Grade Level

Grades 8-10

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