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Consider the function [tex]$f(x)=\left|\frac{-x-3}{2}\right|-1$[/tex].
The piecewise functions that coincide with the given function are
Answer (2)
The function f ( x ) is defined using an absolute value.
We analyze the expression inside the absolute value to determine when it is positive or negative.
Based on the sign of the expression, we split the function into two cases: x ≤ − 3 and -3"> x > − 3 .
We simplify each case to obtain the piecewise function: -3 \end{cases}"> f ( x ) = { 2 − x − 5 , x ≤ − 3 2 x + 1 , x > − 3 .
Explanation
Understanding the Problem We are given the function f ( x ) = 2 − x − 3 − 1 and we want to express it as a piecewise function. The key to this is understanding how the absolute value affects the function's behavior depending on the value of x .
Analyzing the Absolute Value The absolute value function changes its behavior based on the sign of its argument. Specifically, ∣ u ∣ = u if u ≥ 0 , and ∣ u ∣ = − u if u < 0 . In our case, u = 2 − x − 3 . So, we need to determine when 2 − x − 3 is non-negative and when it is negative.
Case 1: x ≤ − 3 Let's find when 2 − x − 3 ≥ 0 . Multiplying both sides by 2 (which doesn't change the inequality since 2 is positive) gives − x − 3 ≥ 0 . Adding x to both sides gives − 3 ≥ x , or x ≤ − 3 . So, when x ≤ − 3 , we have 2 − x − 3 = 2 − x − 3 .
Case 2: -3"> x > − 3 Now, let's find when 2 − x − 3 < 0 . Multiplying both sides by 2 gives − x − 3 < 0 . Adding x to both sides gives − 3 < x , or -3"> x > − 3 . So, when -3"> x > − 3 , we have 2 − x − 3 = − 2 − x − 3 = 2 x + 3 .
Constructing the Piecewise Function Now we can write the piecewise function. When x ≤ − 3 , f ( x ) = 2 − x − 3 − 1 = 2 − x − 3 − 2 = 2 − x − 5 . When -3"> x > − 3 , f ( x ) = 2 x + 3 − 1 = 2 x + 3 − 2 = 2 x + 1 .
Final Piecewise Function Therefore, the piecewise function is -3 \end{cases}"> f ( x ) = { 2 − x − 5 , x ≤ − 3 2 x + 1 , x > − 3 .
Examples
Piecewise functions are useful in modeling situations where the rules change at specific points. For example, consider a cell phone plan where you pay a fixed rate for a certain amount of data, and then a different rate for additional data. This can be modeled using a piecewise function. Similarly, tax brackets, step functions in engineering, and absolute value functions can all be represented using piecewise functions. Understanding piecewise functions helps in analyzing and predicting behavior in these scenarios.
The function f ( x ) = 2 − x − 3 − 1 can be expressed as a piecewise function: -3 \end{cases}"> f ( x ) = { 2 − x − 5 , 2 x + 1 , x ≤ − 3 x > − 3 . This provides a clear interpretation of how the function behaves based on the value of x .
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