Mister Exam
Graphing y = 2-(x-5)/(x^2-5x)
The solution
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x - 5
f(x) = 2 - --------
2
x - 5*x
$$f{\left(x \right)} = - \frac{x - 5}{x^{2} - 5 x} + 2$$
f = -(x - 5)/(x^2 - 5*x) + 2
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{2} = 5$$
$$x_{1} = 0$$
$$x_{2} = 5$$
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:
$$- \frac{x - 5}{x^{2} - 5 x} + 2 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{2}$$
Numerical solution
$$x_{1} = 0.5$$
so we need to solve the equation:
$$- \frac{x - 5}{x^{2} - 5 x} + 2 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{2}$$
Numerical solution
$$x_{1} = 0.5$$
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
$$\frac{\left(5 - 2 x\right) \left(5 - x\right)}{\left(x^{2} - 5 x\right)^{2}} - \frac{1}{x^{2} - 5 x} = 0$$
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
$$\frac{\left(5 - 2 x\right) \left(5 - x\right)}{\left(x^{2} - 5 x\right)^{2}} - \frac{1}{x^{2} - 5 x} = 0$$
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
$$\frac{2 \left(1 + \frac{2 x - 5}{x - 5} - \frac{\left(2 x - 5\right)^{2}}{x \left(x - 5\right)}\right)}{x^{2} \left(x - 5\right)} = 0$$
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
$$\frac{2 \left(1 + \frac{2 x - 5}{x - 5} - \frac{\left(2 x - 5\right)^{2}}{x \left(x - 5\right)}\right)}{x^{2} \left(x - 5\right)} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{2} = 5$$
$$x_{1} = 0$$
$$x_{2} = 5$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(- \frac{x - 5}{x^{2} - 5 x} + 2\right) = 2$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 2$$
$$\lim_{x \to \infty}\left(- \frac{x - 5}{x^{2} - 5 x} + 2\right) = 2$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 2$$
$$\lim_{x \to -\infty}\left(- \frac{x - 5}{x^{2} - 5 x} + 2\right) = 2$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 2$$
$$\lim_{x \to \infty}\left(- \frac{x - 5}{x^{2} - 5 x} + 2\right) = 2$$
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 2 - (x - 5)/(x^2 - 5*x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{- \frac{x - 5}{x^{2} - 5 x} + 2}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{- \frac{x - 5}{x^{2} - 5 x} + 2}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{- \frac{x - 5}{x^{2} - 5 x} + 2}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{- \frac{x - 5}{x^{2} - 5 x} + 2}{x}\right) = 0$$
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:
$$- \frac{x - 5}{x^{2} - 5 x} + 2 = - \frac{- x - 5}{x^{2} + 5 x} + 2$$
- No
$$- \frac{x - 5}{x^{2} - 5 x} + 2 = \frac{- x - 5}{x^{2} + 5 x} - 2$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- \frac{x - 5}{x^{2} - 5 x} + 2 = - \frac{- x - 5}{x^{2} + 5 x} + 2$$
- No
$$- \frac{x - 5}{x^{2} - 5 x} + 2 = \frac{- x - 5}{x^{2} + 5 x} - 2$$
- No
so, the function
not is
neither even, nor odd