Mister Exam
Graphing y = y=(3,5|x|-1)/(|x|-3,5x^2)
The solution
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7*|x|
----- - 1
2
f(x) = ----------
2
7*x
|x| - ----
2
$$f{\left(x \right)} = \frac{\frac{7 \left|{x}\right|}{2} - 1}{- \frac{7 x^{2}}{2} + \left|{x}\right|}$$
f = (7*|x|/2 - 1)/(-7*x^2/2 + |x|)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -0.285714285714286$$
$$x_{2} = 0$$
$$x_{3} = 0.285714285714286$$
$$x_{1} = -0.285714285714286$$
$$x_{2} = 0$$
$$x_{3} = 0.285714285714286$$
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:
$$\frac{\frac{7 \left|{x}\right|}{2} - 1}{- \frac{7 x^{2}}{2} + \left|{x}\right|} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$\frac{\frac{7 \left|{x}\right|}{2} - 1}{- \frac{7 x^{2}}{2} + \left|{x}\right|} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\frac{\left(7 x - \operatorname{sign}{\left(x \right)}\right) \left(\frac{7 \left|{x}\right|}{2} - 1\right)}{\left(- \frac{7 x^{2}}{2} + \left|{x}\right|\right)^{2}} + \frac{7 \operatorname{sign}{\left(x \right)}}{2 \left(- \frac{7 x^{2}}{2} + \left|{x}\right|\right)} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\frac{\left(7 x - \operatorname{sign}{\left(x \right)}\right) \left(\frac{7 \left|{x}\right|}{2} - 1\right)}{\left(- \frac{7 x^{2}}{2} + \left|{x}\right|\right)^{2}} + \frac{7 \operatorname{sign}{\left(x \right)}}{2 \left(- \frac{7 x^{2}}{2} + \left|{x}\right|\right)} = 0$$
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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{2 \left(\frac{14 \left(7 x - \operatorname{sign}{\left(x \right)}\right) \operatorname{sign}{\left(x \right)}}{7 x^{2} - 2 \left|{x}\right|} - 7 \delta\left(x\right) - \frac{\left(7 \left|{x}\right| - 2\right) \left(\frac{4 \left(7 x - \operatorname{sign}{\left(x \right)}\right)^{2}}{7 x^{2} - 2 \left|{x}\right|} + 2 \delta\left(x\right) - 7\right)}{7 x^{2} - 2 \left|{x}\right|}\right)}{7 x^{2} - 2 \left|{x}\right|} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{2 \left(\frac{14 \left(7 x - \operatorname{sign}{\left(x \right)}\right) \operatorname{sign}{\left(x \right)}}{7 x^{2} - 2 \left|{x}\right|} - 7 \delta\left(x\right) - \frac{\left(7 \left|{x}\right| - 2\right) \left(\frac{4 \left(7 x - \operatorname{sign}{\left(x \right)}\right)^{2}}{7 x^{2} - 2 \left|{x}\right|} + 2 \delta\left(x\right) - 7\right)}{7 x^{2} - 2 \left|{x}\right|}\right)}{7 x^{2} - 2 \left|{x}\right|} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = -0.285714285714286$$
$$x_{2} = 0$$
$$x_{3} = 0.285714285714286$$
$$x_{1} = -0.285714285714286$$
$$x_{2} = 0$$
$$x_{3} = 0.285714285714286$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{7 \left|{x}\right|}{2} - 1}{- \frac{7 x^{2}}{2} + \left|{x}\right|}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\frac{7 \left|{x}\right|}{2} - 1}{- \frac{7 x^{2}}{2} + \left|{x}\right|}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\frac{7 \left|{x}\right|}{2} - 1}{- \frac{7 x^{2}}{2} + \left|{x}\right|}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\frac{7 \left|{x}\right|}{2} - 1}{- \frac{7 x^{2}}{2} + \left|{x}\right|}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (7*|x|/2 - 1)/(|x| - 7*x^2/2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{7 \left|{x}\right|}{2} - 1}{x \left(- \frac{7 x^{2}}{2} + \left|{x}\right|\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\frac{7 \left|{x}\right|}{2} - 1}{x \left(- \frac{7 x^{2}}{2} + \left|{x}\right|\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\frac{7 \left|{x}\right|}{2} - 1}{x \left(- \frac{7 x^{2}}{2} + \left|{x}\right|\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\frac{7 \left|{x}\right|}{2} - 1}{x \left(- \frac{7 x^{2}}{2} + \left|{x}\right|\right)}\right) = 0$$
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:
$$\frac{\frac{7 \left|{x}\right|}{2} - 1}{- \frac{7 x^{2}}{2} + \left|{x}\right|} = \frac{\frac{7 \left|{x}\right|}{2} - 1}{- \frac{7 x^{2}}{2} + \left|{x}\right|}$$
- Yes
$$\frac{\frac{7 \left|{x}\right|}{2} - 1}{- \frac{7 x^{2}}{2} + \left|{x}\right|} = - \frac{\frac{7 \left|{x}\right|}{2} - 1}{- \frac{7 x^{2}}{2} + \left|{x}\right|}$$
- No
so, the function
is
even
So, check:
$$\frac{\frac{7 \left|{x}\right|}{2} - 1}{- \frac{7 x^{2}}{2} + \left|{x}\right|} = \frac{\frac{7 \left|{x}\right|}{2} - 1}{- \frac{7 x^{2}}{2} + \left|{x}\right|}$$
- Yes
$$\frac{\frac{7 \left|{x}\right|}{2} - 1}{- \frac{7 x^{2}}{2} + \left|{x}\right|} = - \frac{\frac{7 \left|{x}\right|}{2} - 1}{- \frac{7 x^{2}}{2} + \left|{x}\right|}$$
- No
so, the function
is
even