Mister Exam

Graphing y = x/(9-x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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         x  
f(x) = -----
       9 - x
$$f{\left(x \right)} = \frac{x}{9 - x}$$
f = x/(9 - x)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 9$$
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:
$$\frac{x}{9 - x} = 0$$
Solve this equation
The points of intersection with the axis X:

Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 0$$
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x/(9 - x).
$$\frac{0}{9 - 0}$$
The result:
$$f{\left(0 \right)} = 0$$
The point:
(0, 0)
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
$$\frac{x}{\left(9 - x\right)^{2}} + \frac{1}{9 - x} = 0$$
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
$$\frac{2 \left(- \frac{x}{x - 9} + 1\right)}{\left(x - 9\right)^{2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 9$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{x}{9 - x}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{x \to \infty}\left(\frac{x}{9 - x}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = -1$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x/(9 - x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty} \frac{1}{9 - x} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty} \frac{1}{9 - x} = 0$$
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:
$$\frac{x}{9 - x} = - \frac{x}{x + 9}$$
- No
$$\frac{x}{9 - x} = \frac{x}{x + 9}$$
- No
so, the function
not is
neither even, nor odd