Graphing y = arccos(1/lgx)

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

$$f{\left(x \right)} = \operatorname{acos}{\left(\frac{1}{\log{\left(x \right)}} \right)}$$

f = acos(1/log(x))

The graph of the function

The domain of the function

The points at which the function is not precisely defined:
$$x_{1} = 1$$

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:
$$\operatorname{acos}{\left(\frac{1}{\log{\left(x \right)}} \right)} = 0$$
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 acos(1/log(x)).
$$\operatorname{acos}{\left(\frac{1}{\log{\left(0 \right)}} \right)}$$
The result:
$$f{\left(0 \right)} = \frac{\pi}{2}$$
The point:

(0, pi/2)

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
$$\frac{1}{x \sqrt{1 - \frac{1}{\log{\left(x \right)}^{2}}} \log{\left(x \right)}^{2}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema

Vertical asymptotes

Have:
$$x_{1} = 1$$

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:
$$\operatorname{acos}{\left(\frac{1}{\log{\left(x \right)}} \right)} = \operatorname{acos}{\left(\frac{1}{\log{\left(- x \right)}} \right)}$$
- No
$$\operatorname{acos}{\left(\frac{1}{\log{\left(x \right)}} \right)} = - \operatorname{acos}{\left(\frac{1}{\log{\left(- x \right)}} \right)}$$
- No
so, the function
not is
neither even, nor odd

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Graphing y = arccos(1/lgx)

The graph:

Intersection points:

Piecewise:

The solution