Mister Exam
Graphing y = (sqrt(x)-1)/(lnx)
The solution
You have entered
[src]
___
\/ x - 1
f(x) = ---------
log(x)
$$f{\left(x \right)} = \frac{\sqrt{x} - 1}{\log{\left(x \right)}}$$
f = (sqrt(x) - 1)/log(x)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 1$$
$$x_{1} = 1$$
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:
$$\frac{\sqrt{x} - 1}{\log{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0$$
so we need to solve the equation:
$$\frac{\sqrt{x} - 1}{\log{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0$$
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
$$- \frac{\sqrt{x} - 1}{x \log{\left(x \right)}^{2}} + \frac{1}{2 \sqrt{x} \log{\left(x \right)}} = 0$$
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
$$- \frac{\sqrt{x} - 1}{x \log{\left(x \right)}^{2}} + \frac{1}{2 \sqrt{x} \log{\left(x \right)}} = 0$$
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
$$\frac{\frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \left(\sqrt{x} - 1\right)}{x^{2} \log{\left(x \right)}} - \frac{1}{4 x^{\frac{3}{2}}} - \frac{1}{x^{\frac{3}{2}} \log{\left(x \right)}}}{\log{\left(x \right)}} = 0$$
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
$$\frac{\frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \left(\sqrt{x} - 1\right)}{x^{2} \log{\left(x \right)}} - \frac{1}{4 x^{\frac{3}{2}}} - \frac{1}{x^{\frac{3}{2}} \log{\left(x \right)}}}{\log{\left(x \right)}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 1$$
$$x_{1} = 1$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x} - 1}{\log{\left(x \right)}}\right) = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\sqrt{x} - 1}{\log{\left(x \right)}}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x} - 1}{\log{\left(x \right)}}\right) = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\sqrt{x} - 1}{\log{\left(x \right)}}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (sqrt(x) - 1)/log(x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x} - 1}{x \log{\left(x \right)}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{x} - 1}{x \log{\left(x \right)}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x} - 1}{x \log{\left(x \right)}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{x} - 1}{x \log{\left(x \right)}}\right) = 0$$
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:
$$\frac{\sqrt{x} - 1}{\log{\left(x \right)}} = \frac{\sqrt{- x} - 1}{\log{\left(- x \right)}}$$
- No
$$\frac{\sqrt{x} - 1}{\log{\left(x \right)}} = - \frac{\sqrt{- x} - 1}{\log{\left(- x \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\sqrt{x} - 1}{\log{\left(x \right)}} = \frac{\sqrt{- x} - 1}{\log{\left(- x \right)}}$$
- No
$$\frac{\sqrt{x} - 1}{\log{\left(x \right)}} = - \frac{\sqrt{- x} - 1}{\log{\left(- x \right)}}$$
- No
so, the function
not is
neither even, nor odd