The figure shows the plot of a function over the interval [-4, 4]. Which one of the options given CORRECTLY identifies the function?

Correct Answer:
The correct option is:
$$|2-|x\left|\right|$$

Step-by-Step Explanation:

1. Identify Key Points from the Given Graph:
By observing the plot in the image, we can identify several critical coordinates that the function passes through over the interval $[-4,4]$:
• The y-intercept is at the point $(0,2)$, meaning when $x=0$, $y=2$.
• The x-intercepts are at the points $(-2,0)$ and $(2,0)$, meaning when $x=\pm 2$, $y=0$.
• At the boundaries of the interval, the points are $(-4,2)$ and $(4,2)$, meaning when $x=\pm 4$, $y=2$.

2. Analyze the Function Transformations:
We can reconstruct the target graph using step-by-step graphical transformations as illustrated in the reference image:

Step A: Start with the basic linear graph
Consider the function:
$$y=2-x$$
This is a straight line with a slope of $-1$ and a y-intercept at $(0,2)$.

Step B: Apply absolute value to the input variable $x$
Replacing $x$ with $\left|x\right|$ gives:
$$y=2-\left|x\right|$$
• For $x\ge 0$, this is identical to $y=2-x$.
• Since $\left|x\right|$ is an even function, the graph is symmetric about the y-axis. The portion of the graph for $x<0$ is a reflection of the portion for $x>0$ across the y-axis.
• This results in an inverted V-shaped graph with a peak at $(0,2)$ and x-intercepts at $(-2,0)$ and $(2,0)$.
• For values of $x<-2$ and $x>2$, the y-values become negative (e.g., at $x=\pm 4$, $y=2-4=-2$).

Step C: Apply an outer absolute value to make all outputs non-negative
Taking the absolute value of the entire function gives:
$$y=|2-|x\left|\right|$$
• Any part of the graph of $y=2-\left|x\right|$ that lies below the x-axis (where $y<0$) is reflected across the x-axis to become positive.
• Thus, the negative lines extending downwards for $x<-2$ and $x>2$ are reflected upwards.
• For example, the point $(4,-2)$ becomes $(4,2)$, and the point $(-4,-2)$ becomes $(-4,2)$.
• This transformation results in the characteristic W-shaped graph matching the provided figure exactly.

3. Verification by Substitution:
Let us verify the function $f\left(x\right)=|2-|x\left|\right|$ at the key points:
• $f\left(0\right)=|2-|0\left|\right|=\left|2\right|=2$ (Matches the y-intercept at $(0,2)$)
• $f\left(2\right)=|2-|2\left|\right|=\left|0\right|=0$ (Matches the x-intercept at $(2,0)$)
• $f(-2)=|2-|-2\left|\right|=|2-2|=0$ (Matches the x-intercept at $(-2,0)$)
• $f\left(4\right)=|2-|4\left|\right|=|2-4|=|-2|=2$ (Matches the value at $x=4$)
• $f(-4)=|2-|-4\left|\right|=|2-4|=|-2|=2$ (Matches the value at $x=-4$)