Mister Exam

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Graphing y = |(2x+1)/(x+2)|

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       |2*x + 1|
f(x) = |-------|
       | x + 2 |

f{\left(x \right)} = \left|{\frac{2 x + 1}{x + 2}}\right|

f = Abs((2*x + 1)/(x + 2))

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 x + 1}{x + 2}}\right| = 0

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

Numerical solution

x_{1} = -0.5

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to Abs((2*x + 1)/(x + 2)).

\left|{\frac{0 \cdot 2 + 1}{2}}\right|

The result:

f{\left(0 \right)} = \frac{1}{2}

The point:

(0, 1/2)

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)} =

\left(\frac{2}{x + 2} - \frac{2 x + 1}{\left(x + 2\right)^{2}}\right) \operatorname{sign}{\left(\frac{2 x + 1}{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 x + 1}{x + 2}}\right| = 2

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

y = 2

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

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

y = 2

Inclined asymptotes

Inclined asymptote can be found by calculating the limit of Abs((2*x + 1)/(x + 2)), divided by x at x->+oo and x ->-oo

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

- No

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

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