Mister Exam
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-
How to use it?
- Graphing y =:
- x^3-5/2x^2-2x+3/2
- (x^3)/(4-x^2)
- x³-3x+2
- x^3+6*x^2+9*x+4
-
Identical expressions
- log(x^ two + one)^(one / two)
- logarithm of (x squared plus 1) to the power of (1 divide by 2)
- logarithm of (x to the power of two plus one) to the power of (one divide by two)
- log(x2+1)(1/2)
- logx2+11/2
- log(x²+1)^(1/2)
- log(x to the power of 2+1) to the power of (1/2)
- logx^2+1^1/2
- log(x^2+1)^(1 divide by 2)
-
Similar expressions
- log(x^2-1)^(1/2)
Graphing y = log(x^2+1)^(1/2)
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
$$x_{1} = 0$$
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
$$x_{1} = 0$$
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
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\lim_{x \to -\infty} \sqrt{\log{\left(x^{2} + 1 \right)}} = \infty$$
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
$$\lim_{x \to -\infty}\left(\frac{\sqrt{\log{\left(x^{2} + 1 \right)}}}{x}\right) = 0$$
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
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