Math Problem Statement
f(x)=|x|/x (a)f(2) (b)f(-2) (c)f(x2) (d)f(x-1)
Solution
The function is known as the sign function or the signum function, and it outputs the sign of . The function can be defined piecewise as follows:
\begin{cases} 1 & \text{if } x > 0 \\ -1 & \text{if } x < 0 \\ \text{undefined} & \text{if } x = 0 \end{cases}$$ Let's solve each part: ### (a) $$f(2)$$ Since $$2 > 0$$, we have: $$f(2) = 1$$ ### (b) $$f(-2)$$ Since $$-2 < 0$$, we have: $$f(-2) = -1$$ ### (c) $$f(x^2)$$ Since $$x^2$$ is always non-negative, $$x^2 \geq 0$$, and it is positive when $$x \neq 0$$. Therefore: $$f(x^2) = \begin{cases} 1 & \text{if } x \neq 0 \\ \text{undefined} & \text{if } x = 0 \end{cases}$$ ### (d) $$f(x-1)$$ This function depends on whether $$x-1$$ is positive, negative, or zero: $$f(x-1) = \begin{cases} 1 & \text{if } x > 1 \\ -1 & \text{if } x < 1 \\ \text{undefined} & \text{if } x = 1 \end{cases}$$ Would you like further details on any of these parts or have any additional questions? Here are 5 related questions you might find interesting: 1. How does the sign function behave as $$x$$ approaches zero? 2. Can $$f(x) = \frac{|x|}{x}$$ be differentiated, and if so, where is it differentiable? 3. What is the significance of the function $$f(x) = \frac{|x|}{x}$$ in physics? 4. How does the function $$f(x) = \frac{|x|}{x}$$ compare to the Heaviside step function? 5. How would $$f(x)$$ behave if we added a constant term to $$|x|$$, like $$f(x) = \frac{|x| + c}{x}$$? **Tip:** When analyzing piecewise functions, it's important to consider the definition in each interval and check the behavior at the boundaries.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Absolute value
Piecewise functions
Sign function
Formulas
f(x) = |x|/x
Theorems
-
Suitable Grade Level
High School