Mister Exam
Graphing y = log(x-sqrt(x^2+1))
The solution
You have entered
[src]
/ ________\
| / 2 |
f(x) = log\x - \/ x + 1 /
$$f{\left(x \right)} = \log{\left(x - \sqrt{x^{2} + 1} \right)}$$
f = log(x - sqrt(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:
$$\log{\left(x - \sqrt{x^{2} + 1} \right)} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$\log{\left(x - \sqrt{x^{2} + 1} \right)} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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
$$\frac{- \frac{x}{\sqrt{x^{2} + 1}} + 1}{x - \sqrt{x^{2} + 1}} = 0$$
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
$$\frac{- \frac{x}{\sqrt{x^{2} + 1}} + 1}{x - \sqrt{x^{2} + 1}} = 0$$
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
$$\frac{\frac{\frac{x^{2}}{x^{2} + 1} - 1}{\sqrt{x^{2} + 1}} - \frac{\left(\frac{x}{\sqrt{x^{2} + 1}} - 1\right)^{2}}{x - \sqrt{x^{2} + 1}}}{x - \sqrt{x^{2} + 1}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[0, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\frac{\frac{x^{2}}{x^{2} + 1} - 1}{\sqrt{x^{2} + 1}} - \frac{\left(\frac{x}{\sqrt{x^{2} + 1}} - 1\right)^{2}}{x - \sqrt{x^{2} + 1}}}{x - \sqrt{x^{2} + 1}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[0, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \log{\left(x - \sqrt{x^{2} + 1} \right)} = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \log{\left(x - \sqrt{x^{2} + 1} \right)} = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \log{\left(x - \sqrt{x^{2} + 1} \right)} = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \log{\left(x - \sqrt{x^{2} + 1} \right)} = -\infty$$
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 log(x - sqrt(x^2 + 1)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x - \sqrt{x^{2} + 1} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(x - \sqrt{x^{2} + 1} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x - \sqrt{x^{2} + 1} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(x - \sqrt{x^{2} + 1} \right)}}{x}\right) = 0$$
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:
$$\log{\left(x - \sqrt{x^{2} + 1} \right)} = \log{\left(- x - \sqrt{x^{2} + 1} \right)}$$
- No
$$\log{\left(x - \sqrt{x^{2} + 1} \right)} = - \log{\left(- x - \sqrt{x^{2} + 1} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\log{\left(x - \sqrt{x^{2} + 1} \right)} = \log{\left(- x - \sqrt{x^{2} + 1} \right)}$$
- No
$$\log{\left(x - \sqrt{x^{2} + 1} \right)} = - \log{\left(- x - \sqrt{x^{2} + 1} \right)}$$
- No
so, the function
not is
neither even, nor odd