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sqrt(x^2+1)/x
Plotting a function graph/ sqrt(x^2+1)/x

You entered:

sqrt(x^2+1)/x

What you mean?

sqrt(x^2 + 1)/x
 
sqrt(x*(2 + 1))/x
 
sqrt(x^(2 + 1))/x
 
sqrt(x^2 + 1)/x
 
sqrt(x^2 + 1)/x

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

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          ________
         /  2     
       \/  x  + 1 
f(x) = -----------
            x
f(x)=x2+1xf{\left(x \right)} = \frac{\sqrt{x^{2} + 1}}{x} 
f = sqrt(x^2 + 1)/x
The graph of the function
02468-8-6-4-2-1010-5050
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
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:
x2+1x=0\frac{\sqrt{x^{2} + 1}}{x} = 0
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sqrt(x^2 + 1)/x.
02+10\frac{\sqrt{0^{2} + 1}}{0}
The result:
f(0)=∞~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
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
1x2+1−x2+1x2=0\frac{1}{\sqrt{x^{2} + 1}} - \frac{\sqrt{x^{2} + 1}}{x^{2}} = 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
d2dx2f(x)=0\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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
−x2x2+1−1x2+1−2x2+1+2x2+1x2x=0\frac{- \frac{\frac{x^{2}}{x^{2} + 1} - 1}{\sqrt{x^{2} + 1}} - \frac{2}{\sqrt{x^{2} + 1}} + \frac{2 \sqrt{x^{2} + 1}}{x^{2}}}{x} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
x1=0x_{1} = 0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(x2+1x)=−1\lim_{x \to -\infty}\left(\frac{\sqrt{x^{2} + 1}}{x}\right) = -1
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=−1y = -1
lim⁡x→∞(x2+1x)=1\lim_{x \to \infty}\left(\frac{\sqrt{x^{2} + 1}}{x}\right) = 1
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=1y = 1
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sqrt(x^2 + 1)/x, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(x2+1x2)=0\lim_{x \to -\infty}\left(\frac{\sqrt{x^{2} + 1}}{x^{2}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(x2+1x2)=0\lim_{x \to \infty}\left(\frac{\sqrt{x^{2} + 1}}{x^{2}}\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:
x2+1x=−x2+1x\frac{\sqrt{x^{2} + 1}}{x} = - \frac{\sqrt{x^{2} + 1}}{x}
- No
x2+1x=x2+1x\frac{\sqrt{x^{2} + 1}}{x} = \frac{\sqrt{x^{2} + 1}}{x}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = sqrt(x^2+1)/x
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