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Chapter 1: Problem 73

Graph the function and determine the interval(s) for which \(f(x) \geq 0\). $$ f(x)=-(1+|x|) $$

Short Answer

Expert verified
The function \(f(x) = -(1 + |x|)\) is greater than or equal to 0 when \(x \leq -1\).

Step by step solution

01

Define the function components

The function \(f(x) = -(1 + |x|)\) consists of two function components because of the absolute value: \n a) When \(x < 0\), \(f(x) = -(1 - x)\) \n b) when \(x \geq 0\), \(f(x) = -(1 + x)\
02

Graph the function components

Plot this function on a graph. Notice that these are linear functions with slopes of 1 and -1, respectively, but flipped downwards because of the negative sign.
03

Find the intersection point

To find where the function is 0, set \(f(x)\) to zero and solve for \(x\). It occurs when \(x = -1\). This is where the two components intersect.
04

Determine the interval where f(x) >= 0

From the graph, it can be seen that \(f(x)\) is greater than or equal to zero when \(x \leq -1\).

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