Question

The plot of the function f(x) = ||x| - 1| is

• A.
• B.
• C.
• D.

Answer 1

The Correct Option is B

To analyze the given function \(f(x) = ||x| - 1|\), let's break it down step by step: 

1. The inner absolute function \(|x|\) reflects the input \(x\) so that all outputs are non-negative. This means \(|x|\) is a V-shaped graph centered at the origin.
2. The operation \(|x| - 1\) translates this graph downwards by 1 unit. Therefore, the expression \(|x| - 1\) takes the following form:
   • \( |x| - 1 \) equals \(-1\) when \(-1 \leq x \leq 1\), since at \(x = 0\), \(|0| - 1 = -1\).
   • It equals \(x - 1\) when \(x > 1\) (as \(|x| = x\) for \(x > 0\)).
   • It equals \(-x - 1\) when \(x < -1\) (as \(|x| = -x\) for \(x < 0\)).
3. The outer absolute function \(||x| - 1|\) again makes any resulting negative values positive. Let's see what this transformation does to \(|x| - 1\):
   • For \(|x| - 1 = -1\), the result is \(1\). Thus, for \(-1 \leq x \leq 1\), \(f(x) = 1\).
   • For \(x > 1\), \(|x| - 1 = x - 1\), and since \(x - 1\) is positive, \(f(x) = x - 1\), which linearly increases.
   • For \(x < -1\), \(|x| - 1 = -x - 1\), which is negative, so \(f(x) = x + 1\), which also linearly increases as a reflection in the positive y-axis direction.

Now that we have broken down the transformations, the graph of \(f(x) = ||x| - 1|\) is characterized by a constant value of 1 between \(-1\) and \(1\), and it increases linearly outside this interval.

With these observations, the correct plot for the function is the one that shows these characteristics. Based on the given options, the correct image corresponds to the description above: