Math Problem Statement
Write the absolute value function f(x) = |5x + 4| as a piecewise-defined function with linear parts.
Solution
The problem asks to rewrite the absolute value function as a piecewise-defined function. The key to solving this is to break it into two parts, based on the definition of absolute value:
a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases}$$ For $$f(x) = |5x + 4|$$: 1. The expression $$5x + 4$$ is positive or zero when $$5x + 4 \geq 0$$, i.e., $$x \geq -\frac{4}{5}$$. In this case, $$f(x) = 5x + 4$$. 2. When $$5x + 4 < 0$$, which means $$x < -\frac{4}{5}$$, the absolute value of $$5x + 4$$ becomes $$-(5x + 4)$$. Thus, the piecewise function is: $$f(x) = \begin{cases} 5x + 4 & \text{if } x \geq -\frac{4}{5} \\ -(5x + 4) & \text{if } x < -\frac{4}{5} \end{cases}$$ This matches the structure shown in the boxes on your screen, where you can fill in: $$f(x) = \begin{cases} 5x + 4 & \text{if } x \geq -\frac{4}{5} \\ -(5x + 4) & \text{if } x < -\frac{4}{5} \end{cases}$$ Do you need more details or have any further questions? Here are five related questions to explore: 1. How do you handle the absolute value of more complex functions? 2. What happens when a function inside the absolute value has multiple roots? 3. How would you graph a piecewise function like this one? 4. What is the significance of the point $$x = -\frac{4}{5}$$? 5. How do you integrate piecewise functions? **Tip:** Always identify where the expression inside the absolute value changes sign to set up your piecewise function correctly!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Piecewise Functions
Absolute Value Functions
Formulas
|a| = a if a ≥ 0; -a if a < 0
f(x) = |5x + 4| rewritten as a piecewise function
Theorems
Absolute value definition
Piecewise function construction
Suitable Grade Level
Grades 8-10
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