Understanding Piecewise Function f(x) = -12x, -4, √x

Math Problem Statement

f(x)=⎧⎪ ⎪⎨⎪ ⎪⎩−12xifx<−6−4if−6≤x<0√xifx≥0

The function $f(x)$ you provided is a piecewise function defined as:

\begin{cases} -12x & \text{if } x < -6 \\ -4 & \text{if } -6 \leq x < 0 \\ \sqrt{x} & \text{if } x \geq 0 \end{cases}$$ This means that the value of the function depends on the value of $$x$$ in different intervals. Let's break it down: 1. **For $$x < -6$$**: The function is $$f(x) = -12x$$. 2. **For $$-6 \leq x < 0$$**: The function is $$f(x) = -4$$. 3. **For $$x \geq 0$$**: The function is $$f(x) = \sqrt{x}$$, which means the square root of $$x$$. Would you like to evaluate this function at certain values of $$x$$ or explore its properties further? Here are some follow-up questions to help expand the discussion: 1. How would you find the limit of $$f(x)$$ as $$x \to -6$$? 2. What is the domain and range of this piecewise function? 3. How would you calculate the derivative of $$f(x)$$ in each piecewise segment? 4. What happens to $$f(x)$$ as $$x$$ approaches 0 from the left and right? 5. How do you sketch the graph of this piecewise function? **Tip**: When working with piecewise functions, it's crucial to ensure the function is defined for all ranges of $$x$$ and that the conditions of the intervals are well understood.

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Piecewise Functions
Square Root Functions
Limits

f(x) = -12x for x < -6
f(x) = -4 for -6 ≤ x < 0
f(x) = √x for x ≥ 0

Continuity of Piecewise Functions
Limits of Piecewise Functions

Grades 11-12

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