Math Problem Statement

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

Solution

The function you've given is a piecewise function, and we'll find the points for each segment.

The function is 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}$$ Let's break it down: ### 1. For $$x < -6$$: The function is $$f(x) = -12x$$. This is a straight line with slope -12. For any value of $$x$$ less than -6, you can find the corresponding $$f(x)$$ by plugging $$x$$ into this equation. - Example points for this section: - When $$x = -7$$, $$f(-7) = -12(-7) = 84$$ - When $$x = -8$$, $$f(-8) = -12(-8) = 96$$ ### 2. For $$-6 \leq x < 0$$: The function is constant, $$f(x) = -4$$. So, for all values of $$x$$ between -6 and 0, $$f(x)$$ will always be -4. - Example points: - When $$x = -6$$, $$f(-6) = -4$$ - When $$x = -2$$, $$f(-2) = -4$$ - When $$x = -1$$, $$f(-1) = -4$$ ### 3. For $$x \geq 0$$: The function is $$f(x) = \sqrt{x}$$, which is the square root of $$x$$. For any value of $$x$$ greater than or equal to 0, $$f(x)$$ is the square root of $$x$$. - Example points: - When $$x = 0$$, $$f(0) = \sqrt{0} = 0$$ - When $$x = 1$$, $$f(1) = \sqrt{1} = 1$$ - When $$x = 4$$, $$f(4) = \sqrt{4} = 2$$ ### Summary of example points: - For $$x = -7$$, $$f(x) = 84$$ - For $$x = -6$$, $$f(x) = -4$$ - For $$x = -2$$, $$f(x) = -4$$ - For $$x = -1$$, $$f(x) = -4$$ - For $$x = 0$$, $$f(x) = 0$$ - For $$x = 1$$, $$f(x) = 1$$ - For $$x = 4$$, $$f(x) = 2$$ Let me know if you'd like more points or further explanation! ### Further questions to explore: 1. How do the function values behave at the boundary points like $$x = -6$$ and $$x = 0$$? 2. What happens to the slope of the function for $$x < -6$$ compared to other intervals? 3. Can you calculate the derivative of each part of the piecewise function? 4. What is the continuity of the function at $$x = -6$$ and $$x = 0$$? 5. How would you graph this piecewise function by hand? ### Tip: Piecewise functions can often be graphed by first sketching each section individually and then connecting them smoothly where needed!

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Linear Functions
Square Root Function

Formulas

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

Theorems

Piecewise function evaluation
Function behavior at boundary points

Suitable Grade Level

Grades 9-12