Math Problem Statement
Write the piecewise function that is equivalent to the absolute value function f(x)=|x+8|
Solution
To write the piecewise function equivalent to the absolute value function , we need to express it in terms of two cases, depending on whether is positive or negative.
The absolute value function can be written as:
\begin{cases} x + 8 & \text{if } x + 8 \geq 0 \\ -(x + 8) & \text{if } x + 8 < 0 \end{cases}$$ Now, let’s simplify the conditions: 1. For $$x + 8 \geq 0$$: - This implies $$x \geq -8$$. - So, $$f(x) = x + 8$$ when $$x \geq -8$$. 2. For $$x + 8 < 0$$: - This implies $$x < -8$$. - So, $$f(x) = -(x + 8) = -x - 8$$ when $$x < -8$$. Therefore, the piecewise function is: $$f(x) = \begin{cases} x + 8 & \text{if } x \geq -8 \\ -x - 8 & \text{if } x < -8 \end{cases}$$ Would you like further details or clarification on any steps? Here are five related questions for further exploration: 1. How do we write piecewise functions for other absolute value functions, like $$|x - 5|$$? 2. How can we graph this piecewise function to visualize it? 3. How do we handle absolute value functions with expressions like $$f(x) = |2x - 3|$$? 4. What happens to the piecewise function if we add a constant outside the absolute value, such as $$f(x) = |x + 8| + 3$$? 5. How do we solve equations involving $$|x + 8| = 4$$ using this piecewise function? **Tip:** Always check the domain of each piece when defining piecewise functions for absolute values, as it helps ensure correct conditions for each segment.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Absolute Value Functions
Piecewise Functions
Formulas
f(x) = |x + 8| = { x + 8 if x ≥ -8, -x - 8 if x < -8 }
Theorems
Absolute Value Definition
Piecewise Function Definition
Suitable Grade Level
Grades 8-10