Mister Exam
Graphing y = (x-4)/(x-6)
The solution
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x - 4
f(x) = -----
x - 6
$$f{\left(x \right)} = \frac{x - 4}{x - 6}$$
f = (x - 4)/(x - 6)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 6$$
$$x_{1} = 6$$
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:
$$\frac{x - 4}{x - 6} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 4$$
Numerical solution
$$x_{1} = 4$$
so we need to solve the equation:
$$\frac{x - 4}{x - 6} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 4$$
Numerical solution
$$x_{1} = 4$$
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
$$\frac{1}{x - 6} - \frac{x - 4}{\left(x - 6\right)^{2}} = 0$$
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
$$\frac{1}{x - 6} - \frac{x - 4}{\left(x - 6\right)^{2}} = 0$$
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
$$\frac{2 \left(-1 + \frac{x - 4}{x - 6}\right)}{\left(x - 6\right)^{2}} = 0$$
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
$$\frac{2 \left(-1 + \frac{x - 4}{x - 6}\right)}{\left(x - 6\right)^{2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 6$$
$$x_{1} = 6$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{x - 4}{x - 6}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty}\left(\frac{x - 4}{x - 6}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
$$\lim_{x \to -\infty}\left(\frac{x - 4}{x - 6}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty}\left(\frac{x - 4}{x - 6}\right) = 1$$
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 - 4)/(x - 6), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{x - 4}{x \left(x - 6\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x - 4}{x \left(x - 6\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{x - 4}{x \left(x - 6\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x - 4}{x \left(x - 6\right)}\right) = 0$$
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:
$$\frac{x - 4}{x - 6} = \frac{- x - 4}{- x - 6}$$
- No
$$\frac{x - 4}{x - 6} = - \frac{- x - 4}{- x - 6}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{x - 4}{x - 6} = \frac{- x - 4}{- x - 6}$$
- No
$$\frac{x - 4}{x - 6} = - \frac{- x - 4}{- x - 6}$$
- No
so, the function
not is
neither even, nor odd