Question
Let \( f(x)=|x|-6 \) Use the definition of the absolute value to expand and write the function as a piecewise define function, Choose the most correct answer, (A) \( f(x)=|x|-6=\left\{\begin{array}{ll}x-6 & \text { if } x \geq 6 \\ -x-6 & \text { if } x<6\end{array}\right. \) (B) \( f(x)=|x|-6=\left\{\begin{array}{ll}x-6 & \text { if } x \geq 6 \\ -x+6 & \text { if } x<6\end{array}\right. \) (C) \( f(x)=|x|-6=\left\{\begin{array}{ll}x-6 & \text { if } x \geq 0 \\ -x+6 & \text { if } x<0\end{array}\right. \) (D) \( f(x)=|x|-6=\left\{\begin{array}{ll}x-6 & \text { if } x \geq 0 \\ -x-6 & \text { if } x<0\end{array}\right. \)
Ask by Barnett Guerrero. in South Africa
Feb 21,2025
Upstudy AI Solution
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Answer
The correct answer is (D) \( f(x) = |x| - 6 = \left\{ \begin{array}{ll} x - 6 & \text{if } x \geq 0 \\ -x - 6 & \text{if } x < 0 \end{array} \right. \)
Solution
We're given f(x) = |x| - 6. The definition of absolute value states that for any real number x:
• |x| = x if x ≥ 0
• |x| = -x if x < 0
Thus, we can rewrite f(x) as follows:
• If x ≥ 0, then f(x) = x - 6.
• If x < 0, then f(x) = (-x) - 6 = -x - 6.
The piecewise function is therefore:
f(x) = { x - 6 if x ≥ 0
-x - 6 if x < 0 }
Comparing with the provided answers, this matches option (D).
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Simplify this solution Beyond the Answer
To expand \( f(x) = |x| - 6 \) using the definition of the absolute value, we consider two cases: when \( x \geq 0 \) and when \( x < 0 \). 1. **If \( x \geq 0 \)**, then \( |x| = x \). Thus, \( f(x) = x - 6 \). 2. **If \( x < 0 \)**, then \( |x| = -x \). Therefore, \( f(x) = -x - 6 \). Combining these cases, we get: \[ f(x) = \left\{ \begin{array}{ll} x - 6 & \text{if } x \geq 0 \\ -x - 6 & \text{if } x < 0 \end{array} \right. \] Thus, the correct piecewise definition of the function is: (D) \( f(x)=|x|-6=\left\{\begin{array}{ll}x-6 & \text { if } x \geq 0 \\ -x-6 & \text { if } x<0\end{array}\right. \)
