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log(x^2+1)^(1/2)

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  • Identical expressions

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  • log(x^2+1)^(1 divide by 2)

  • Similar expressions

  • log(x^2-1)^(1/2)
Plotting a function graph/ log(x^2+1)^(1/2)

Graphing y = log(x^2+1)^(1/2)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          _____________
         /    / 2    \ 
f(x) = \/  log\x  + 1/
$$f{\left(x \right)} = \sqrt{\log{\left(x^{2} + 1 \right)}}$$ 
f = sqrt(log(x^2 + 1))
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:
$$\sqrt{\log{\left(x^{2} + 1 \right)}} = 0$$
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 sqrt(log(x^2 + 1)).
$$\sqrt{\log{\left(0^{2} + 1 \right)}}$$
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
$$\frac{x}{\left(x^{2} + 1\right) \sqrt{\log{\left(x^{2} + 1 \right)}}} = 0$$
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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{- \frac{2 x^{2}}{x^{2} + 1} - \frac{x^{2}}{\left(x^{2} + 1\right) \log{\left(x^{2} + 1 \right)}} + 1}{\left(x^{2} + 1\right) \sqrt{\log{\left(x^{2} + 1 \right)}}} = 0$$
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
$$\lim_{x \to -\infty} \sqrt{\log{\left(x^{2} + 1 \right)}} = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \sqrt{\log{\left(x^{2} + 1 \right)}} = \infty$$
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(log(x^2 + 1)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt{\log{\left(x^{2} + 1 \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{\log{\left(x^{2} + 1 \right)}}}{x}\right) = 0$$
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:
$$\sqrt{\log{\left(x^{2} + 1 \right)}} = \sqrt{\log{\left(x^{2} + 1 \right)}}$$
- Yes
$$\sqrt{\log{\left(x^{2} + 1 \right)}} = - \sqrt{\log{\left(x^{2} + 1 \right)}}$$
- No
so, the function
is
even
The graph
Graphing y = log(x^2+1)^(1/2)
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