Mister Exam

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Graphing y = sqrt(sqr(log(x)-1))

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f(x) = \/  (log(x) - 1)

f{\left(x \right)} = \sqrt{\left(\log{\left(x \right)} - 1\right)^{2}}

f = sqrt((log(x) - 1)^2)

The graph of the function

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:

\sqrt{\left(\log{\left(x \right)} - 1\right)^{2}} = 0

Solve this equation
The points of intersection with the axis X:

Analytical solution

x_{1} = e

Numerical solution

x_{1} = 2.71828182845905

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to sqrt((log(x) - 1)^2).

\sqrt{\left(\log{\left(0 \right)} - 1\right)^{2}}

The result:

f{\left(0 \right)} = \tilde{\infty}

sof doesn't intersect Y

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)} =

\frac{\sqrt{\left(\log{\left(x \right)} - 1\right)^{2}}}{x \left(\log{\left(x \right)} - 1\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)} =

- \frac{\sqrt{\left(\log{\left(x \right)} - 1\right)^{2}}}{x^{2} \left(\log{\left(x \right)} - 1\right)} = 0

Solve this equation
Solutions are not found,
maybe, the function has no inflections

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} \sqrt{\left(\log{\left(x \right)} - 1\right)^{2}} = \infty

Let's take the limit
so,
horizontal asymptote on the left doesn’t exist

\lim_{x \to \infty} \sqrt{\left(\log{\left(x \right)} - 1\right)^{2}} = \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((log(x) - 1)^2), divided by x at x->+oo and x ->-oo

\lim_{x \to -\infty}\left(\frac{\sqrt{\left(\log{\left(x \right)} - 1\right)^{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{\sqrt{\left(\log{\left(x \right)} - 1\right)^{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:

\sqrt{\left(\log{\left(x \right)} - 1\right)^{2}} = \sqrt{\left(\log{\left(- x \right)} - 1\right)^{2}}

- No

\sqrt{\left(\log{\left(x \right)} - 1\right)^{2}} = - \sqrt{\left(\log{\left(- x \right)} - 1\right)^{2}}

- No
so, the function
not is
neither even, nor odd