Mister Exam

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Graphing y = (abs((2-3*x)/(2-x)))

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       |2 - 3*x|
f(x) = |-------|
       | 2 - x |

f{\left(x \right)} = \left|{\frac{2 - 3 x}{2 - x}}\right|

f = Abs((2 - 3*x)/(2 - x))

The graph of the function

The domain of the function

The points at which the function is not precisely defined:

x_{1} = 2

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|{\frac{2 - 3 x}{2 - x}}\right| = 0

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

Numerical solution

x_{1} = 0.666666666666667

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to Abs((2 - 3*x)/(2 - x)).

\left|{\frac{2 - 0}{2 - 0}}\right|

The result:

f{\left(0 \right)} = 1

The point:

(0, 1)

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)} =

\frac{\left(2 - 3 x\right) \left(x - 2\right) \left(\frac{2 - 3 x}{\left(2 - x\right)^{2}} - \frac{3}{2 - x}\right) \operatorname{sign}{\left(\frac{3 x - 2}{x - 2} \right)}}{\left(2 - x\right) \left(3 x - 2\right)} = 0

Solve this equation
Solutions are not found,
function may have no extrema

Vertical asymptotes

Have:

x_{1} = 2

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|{\frac{2 - 3 x}{2 - x}}\right| = 3

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

y = 3

\lim_{x \to \infty} \left|{\frac{2 - 3 x}{2 - x}}\right| = 3

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

y = 3

Inclined asymptotes

Inclined asymptote can be found by calculating the limit of Abs((2 - 3*x)/(2 - x)), divided by x at x->+oo and x ->-oo

\lim_{x \to -\infty}\left(\frac{\left|{\frac{2 - 3 x}{2 - x}}\right|}{x}\right) = 0

Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right

\lim_{x \to \infty}\left(\frac{\left|{\frac{2 - 3 x}{2 - x}}\right|}{x}\right) = 0

Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left

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|{\frac{2 - 3 x}{2 - x}}\right| = \left|{\frac{3 x + 2}{x + 2}}\right|

- No

\left|{\frac{2 - 3 x}{2 - x}}\right| = - \left|{\frac{3 x + 2}{x + 2}}\right|

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