Math Problem Statement
F(x)=2|x|
Solution
The function involves an absolute value. Here's a breakdown of its properties:
1. Definition of :
The absolute value function, , makes all input values non-negative. Therefore:
\begin{cases} 2x & \text{if } x \geq 0 \\ -2x & \text{if } x < 0 \end{cases}$$ ### 2. **Graph of $$F(x)$$:** - For $$x \geq 0$$, the graph is a straight line with slope $$2$$ (passes through the origin). - For $$x < 0$$, the graph is a straight line with slope $$-2$$ (also passes through the origin). The graph forms a "V" shape, symmetric about the $$y$$-axis. ### 3. **Key Properties:** - **Domain:** $$(-\infty, \infty)$$ (all real numbers). - **Range:** $$[0, \infty)$$ (outputs are non-negative because of the absolute value). - **Continuity:** The function is continuous everywhere (no breaks in the graph). - **Differentiability:** The function is differentiable everywhere except at $$x = 0$$, where the slope abruptly changes. ### 4. **Derivative of $$F(x)$$:** The derivative depends on the domain: $$F'(x) = \begin{cases} 2 & \text{if } x > 0 \\ -2 & \text{if } x < 0 \\ \text{undefined} & \text{at } x = 0 \end{cases}$$ ### 5. **Applications:** This function is used to model situations where the magnitude of a value matters regardless of direction, like distance or cost. --- Would you like a graphical representation of this function or further analysis? ### 5 Related Questions: 1. How does $$F(x) = 2|x|$$ compare to $$G(x) = |x|$$? 2. What are the limits of $$F(x)$$ as $$x \to \infty$$ and $$x \to -\infty$$? 3. What is the second derivative of $$F(x)$$? 4. Can $$F(x)$$ be expressed as a piecewise linear function? If so, how does this affect its application? 5. How does the absolute value impact the symmetry of the function? ### Tip: When dealing with absolute value functions, always consider splitting the domain into $$x \geq 0$$ and $$x < 0$$ for easier analysis.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Absolute Value
Piecewise Functions
Graphing Linear Functions
Differentiation
Formulas
F(x) = 2|x| = { 2x if x >= 0, -2x if x < 0 }
Derivative: F'(x) = { 2 if x > 0, -2 if x < 0, undefined at x = 0 }
Theorems
Continuity of Absolute Value Functions
Differentiability of Piecewise Functions
Suitable Grade Level
Grades 10-12
Related Recommendation
Understanding Absolute Value Function: f(x) = |x + 1|
Behavior of Absolute Value Function 2|x + 2| + 1 As x Increases and Decreases
Domain and Range of the Absolute Value Function f(x) = 2|x+2| + 1
Analyzing and Reflecting Absolute Value Function f(x) = 2|x|
Understanding the Transformation, Domain, and Range of f(x) = |x| + 2