Math Problem Statement

Graph the piecewise function f(x) = |x| for x < 2 and f(x) = -x + 4 for x ≥ 2

Solution

Let's analyze the function f(x)f(x) defined piecewise:

\begin{cases} |x| & \text{for } x < 2 \\ -x + 4 & \text{for } x \geq 2 \end{cases}$$ ### Steps to graph the function: 1. **For $$x < 2$$, the function is $$f(x) = |x|$$:** - The absolute value function $$|x|$$ is V-shaped, with a vertex at $$x = 0$$. - For $$x < 2$$, plot the points: - $$f(-3) = 3$$ - $$f(-2) = 2$$ - $$f(0) = 0$$ - $$f(1) = 1$$ - As $$x$$ approaches 2 from the left, $$f(1.5) = 1.5$$ and $$f(2) = 2$$, but this is the boundary for the next function piece. 2. **For $$x \geq 2$$, the function is $$f(x) = -x + 4$$:** - This is a linear function with a slope of -1 and a y-intercept of 4. - Plot points for $$x \geq 2$$: - At $$x = 2$$, $$f(2) = 2$$ (continuous from the previous piece). - $$f(3) = -3 + 4 = 1$$ - $$f(4) = -4 + 4 = 0$$ - $$f(5) = -5 + 4 = -1$$ Now graph the two parts: - For $$x < 2$$, the function is the absolute value graph, with the point $$(2, 2)$$ included. - For $$x \geq 2$$, graph the line $$-x + 4$$, starting from $$(2, 2)$$. Would you like further clarification or help with plotting? Here are some questions for further practice: 1. What is the domain and range of this piecewise function? 2. Is the function continuous at $$x = 2$$? 3. How would the graph change if the linear part was $$-x + 5$$? 4. How do we define the slope for $$f(x) = -x + 4$$? 5. What is the behavior of the absolute value function as $$x \to -\infty$$? **Tip:** When graphing piecewise functions, always check for continuity and slope at the boundary points.

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Math Problem Analysis

Mathematical Concepts

Piecewise functions
Absolute value functions
Linear functions
Graphing functions

Formulas

f(x) = |x| for x < 2
f(x) = -x + 4 for x ≥ 2

Theorems

Continuity at x = 2
Piecewise function continuity and slope analysis

Suitable Grade Level

Grades 9-12