Analyzing a Piecewise Function: Linear and Cosine Segments

Math Problem Statement

g(x)=\begin{cases} \dfrac x3-2&\text{for }0<x<6 \\ \text{cos}(x\cdot\pi)&\text{for }6\leq x\leq10 \end{cases}

The given function $g(x)$ is a piecewise-defined function, meaning it has different expressions for different intervals of $x$ . Let's break it down:

\begin{cases} \dfrac{x}{3} - 2 & \text{for } 0 < x < 6 \\ \cos(x \cdot \pi) & \text{for } 6 \leq x \leq 10 \end{cases}$$ ### Explanation: 1. **For $$0 < x < 6$$:** - The function is linear: $$g(x) = \dfrac{x}{3} - 2$$. - This is a straight line with slope $$\frac{1}{3}$$ and y-intercept $$-2$$. - It only applies for $$x$$ values strictly between 0 and 6 (not including the endpoints). 2. **For $$6 \leq x \leq 10$$:** - The function is $$g(x) = \cos(x \cdot \pi)$$, which is a cosine function scaled by $$\pi$$. - This applies for $$x$$ values from 6 to 10 (including the endpoints 6 and 10). ### Points of Interest: - The function transitions from a linear function to a trigonometric one at $$x = 6$$. - At $$x = 6$$, you may want to check for continuity: - From the first piece, as $$x \to 6^{-}$$, $$g(x) = \frac{6}{3} - 2 = 0$$. - At $$x = 6$$ in the second piece, $$g(6) = \cos(6\pi) = 1$$. - So, $$g(x)$$ is **not continuous** at $$x = 6$$. Would you like to explore details like graphing, continuity, or limits for this function? ### Follow-Up Questions: 1. What is the value of $$g(x)$$ when $$x = 8$$? 2. Is the function continuous at $$x = 6$$? 3. Can you find the derivative of $$g(x)$$ for $$0 < x < 6$$? 4. What is the value of $$\cos(7 \pi)$$ in the second piece? 5. How does the function behave as $$x \to 0^+$$? ### Tip: For piecewise functions, always check the transition points for continuity and differentiability.

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Piecewise Functions
Linear Equations
Trigonometric Functions
Continuity

g(x) = \dfrac{x}{3} - 2
g(x) = \cos(x \cdot \pi)

Continuity at a Point

Grades 10-12

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