Graphing y = |x-1|+|x-1|

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f(x) = |x - 1| + |x - 1|

f{\left(x \right)} = \left|{x - 1}\right| + \left|{x - 1}\right|

f = |x - 1| + |x - 1|

The graph of the function

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:

\left|{x - 1}\right| + \left|{x - 1}\right| = 0

Solve this equation
The points of intersection with the axis X:

Analytical solution

x_{1} = 1

Numerical solution

x_{1} = 1

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to |x - 1| + |x - 1|.

\left|{-1}\right| + \left|{-1}\right|

The result:

f{\left(0 \right)} = 2

The point:

(0, 2)

Extrema of the function

In order to find the extrema, we need to solve the equation

\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:

\frac{d}{d x} f{\left(x \right)} =

2 \operatorname{sign}{\left(x - 1 \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

\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:

\frac{d^{2}}{d x^{2}} f{\left(x \right)} =

4 \delta\left(x - 1\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 \to -\infty}\left(\left|{x - 1}\right| + \left|{x - 1}\right|\right) = \infty

Let's take the limit
so,
horizontal asymptote on the left doesn’t exist

\lim_{x \to \infty}\left(\left|{x - 1}\right| + \left|{x - 1}\right|\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 |x - 1| + |x - 1|, divided by x at x->+oo and x ->-oo

\lim_{x \to -\infty}\left(\frac{\left|{x - 1}\right| + \left|{x - 1}\right|}{x}\right) = -2

Let's take the limit
so,
inclined asymptote equation on the left:

y = - 2 x

\lim_{x \to \infty}\left(\frac{\left|{x - 1}\right| + \left|{x - 1}\right|}{x}\right) = 2

Let's take the limit
so,
inclined asymptote equation on the right:

y = 2 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:

\left|{x - 1}\right| + \left|{x - 1}\right| = 2 \left|{x + 1}\right|

- No

\left|{x - 1}\right| + \left|{x - 1}\right| = - 2 \left|{x + 1}\right|

- No
so, the function
not is
neither even, nor odd