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Plotting a function graph/ ||x-3|-2|

Graphing y = ||x-3|-2|

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = ||x - 3| - 2|
f(x)=∣∣x−3∣−2∣f{\left(x \right)} = \left|{\left|{x - 3}\right| - 2}\right| 
f = Abs(|x - 3| - 2)
The graph of the function
02468-8-6-4-2-1010020
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
∣∣x−3∣−2∣=0\left|{\left|{x - 3}\right| - 2}\right| = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=1x_{1} = 1
x2=5x_{2} = 5
Numerical solution
x1=1x_{1} = 1
x2=5x_{2} = 5
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to Abs(|x - 3| - 2).
∣−2+∣−3∣∣\left|{-2 + \left|{-3}\right|}\right|
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
sign⁡(x−3)sign⁡(∣x−3∣−2)=0\operatorname{sign}{\left(x - 3 \right)} \operatorname{sign}{\left(\left|{x - 3}\right| - 2 \right)} = 0
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
2(δ(x−3)sign⁡(∣x−3∣−2)+δ(∣x−3∣−2)sign⁡2(x−3))=02 \left(\delta\left(x - 3\right) \operatorname{sign}{\left(\left|{x - 3}\right| - 2 \right)} + \delta\left(\left|{x - 3}\right| - 2\right) \operatorname{sign}^{2}{\left(x - 3 \right)}\right) = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞∣∣x−3∣−2∣=∞\lim_{x \to -\infty} \left|{\left|{x - 3}\right| - 2}\right| = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞∣∣x−3∣−2∣=∞\lim_{x \to \infty} \left|{\left|{x - 3}\right| - 2}\right| = \infty
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of Abs(|x - 3| - 2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(∣∣x−3∣−2∣x)=−1\lim_{x \to -\infty}\left(\frac{\left|{\left|{x - 3}\right| - 2}\right|}{x}\right) = -1
Let's take the limit
so,
inclined asymptote equation on the left:
y=−xy = - x
lim⁡x→∞(∣∣x−3∣−2∣x)=1\lim_{x \to \infty}\left(\frac{\left|{\left|{x - 3}\right| - 2}\right|}{x}\right) = 1
Let's take the limit
so,
inclined asymptote equation on the right:
y=xy = x
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
∣∣x−3∣−2∣=∣∣x+3∣−2∣\left|{\left|{x - 3}\right| - 2}\right| = \left|{\left|{x + 3}\right| - 2}\right|
- No
∣∣x−3∣−2∣=−∣∣x+3∣−2∣\left|{\left|{x - 3}\right| - 2}\right| = - \left|{\left|{x + 3}\right| - 2}\right|
- No
so, the function
not is
neither even, nor odd
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