Math Problem Statement

the updated piecewise function looks like this:

f(x)={−1,if x≤−12x+1,if −1<x≤12,if x>1f(x)=⎩⎨⎧​−1,2x+1,2,​if x≤−1if −1<x≤1if x>1​

Solution

The piecewise function you provided can be clarified as follows:

\begin{cases} -1, & \text{if } x \leq -1 \\ 2x + 1, & \text{if } -1 < x \leq 1 \\ 2, & \text{if } x > 1 \end{cases}$$ ### Explanation: - **For $$x \leq -1$$**: The value of the function is $$-1$$. - **For $$-1 < x \leq 1$$**: The function is linear, with the expression $$f(x) = 2x + 1$$. - **For $$x > 1$$**: The function takes a constant value of 2. Would you like further details or visual representation of this function? ### Related Questions: 1. What is the domain of this piecewise function? 2. How would you find the limits of $$f(x)$$ as $$x \to -1$$ and $$x \to 1$$? 3. What is the range of the function? 4. How does the function behave around the discontinuities at $$x = -1$$ and $$x = 1$$? 5. How would you graph this piecewise function? **Tip**: For piecewise functions, always check continuity and differentiability at the points where the function changes definition!

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Linear Functions
Inequalities
Continuity

Formulas

f(x) = -1, if x ≤ -1
f(x) = 2x + 1, if -1 < x ≤ 1
f(x) = 2, if x > 1

Theorems

Continuity and Discontinuity at the boundaries
Linear Equation Properties

Suitable Grade Level

Grades 10-12