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

You entered:

(x^2+1)/x^2

What you mean?

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

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

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The graph:

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Intersection points:

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Piecewise:

The solution

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        2    
       x  + 1
f(x) = ------
          2  
         x
f(x)=x2+1x2f{\left(x \right)} = \frac{x^{2} + 1}{x^{2}} 
f = (x^2 + 1)/(x^2)
The graph of the function
-1.0-0.8-0.6-0.4-0.21.00.00.20.40.60.8050000
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+1x2=0\frac{x^{2} + 1}{x^{2}} = 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 (x^2 + 1)/(x^2).
02+102\frac{0^{2} + 1}{0^{2}}
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
2xx2−2(x2+1)x3=0\frac{2 x}{x^{2}} - \frac{2 \left(x^{2} + 1\right)}{x^{3}} = 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
6(−1+x2+1x2)x2=0\frac{6 \left(-1 + \frac{x^{2} + 1}{x^{2}}\right)}{x^{2}} = 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+1x2)=1\lim_{x \to -\infty}\left(\frac{x^{2} + 1}{x^{2}}\right) = 1
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=1y = 1
lim⁡x→∞(x2+1x2)=1\lim_{x \to \infty}\left(\frac{x^{2} + 1}{x^{2}}\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 (x^2 + 1)/(x^2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(x2+1xx2)=0\lim_{x \to -\infty}\left(\frac{x^{2} + 1}{x x^{2}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(x2+1xx2)=0\lim_{x \to \infty}\left(\frac{x^{2} + 1}{x 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+1x2=x2+1x2\frac{x^{2} + 1}{x^{2}} = \frac{x^{2} + 1}{x^{2}}
- Yes
x2+1x2=−x2+1x2\frac{x^{2} + 1}{x^{2}} = - \frac{x^{2} + 1}{x^{2}}
- No
so, the function
is
even
The graph
Graphing y = (x^2+1)/x^2
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