Mister Exam
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-
How to use it?
- Graphing y =:
- x^3+6*x^2+9*x+4
- x^3+4x
- (x^2+x-5)/(x-2)
-
|x^2+x-2|
- Integral of d{x}:
-
(x^2+1)/x^2
- Derivative of:
-
(x^2+1)/x^2
-
Identical expressions
- (x^ two + one)/x^ two
- (x squared plus 1) divide by x squared
- (x to the power of two plus one) divide by x to the power of two
- (x2+1)/x2
- x2+1/x2
- (x²+1)/x²
- (x to the power of 2+1)/x to the power of 2
- x^2+1/x^2
- (x^2+1) divide by x^2
-
Similar expressions
- (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
You entered:
(x^2+1)/x^2
What you mean?
Graphing y = (x^2+1)/x^2
The solution
You have entered
[src]
2
x + 1
f(x) = ------
2
x
$$f{\left(x \right)} = \frac{x^{2} + 1}{x^{2}}$$
f = (x^2 + 1)/(x^2)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\frac{x^{2} + 1}{x^{2}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$\frac{x^{2} + 1}{x^{2}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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
$$\frac{2 x}{x^{2}} - \frac{2 \left(x^{2} + 1\right)}{x^{3}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{2 x}{x^{2}} - \frac{2 \left(x^{2} + 1\right)}{x^{3}} = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{6 \left(-1 + \frac{x^{2} + 1}{x^{2}}\right)}{x^{2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{6 \left(-1 + \frac{x^{2} + 1}{x^{2}}\right)}{x^{2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{x^{2} + 1}{x^{2}}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty}\left(\frac{x^{2} + 1}{x^{2}}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
$$\lim_{x \to -\infty}\left(\frac{x^{2} + 1}{x^{2}}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty}\left(\frac{x^{2} + 1}{x^{2}}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
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
$$\lim_{x \to -\infty}\left(\frac{x^{2} + 1}{x x^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x^{2} + 1}{x x^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{x^{2} + 1}{x x^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x^{2} + 1}{x x^{2}}\right) = 0$$
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:
$$\frac{x^{2} + 1}{x^{2}} = \frac{x^{2} + 1}{x^{2}}$$
- Yes
$$\frac{x^{2} + 1}{x^{2}} = - \frac{x^{2} + 1}{x^{2}}$$
- No
so, the function
is
even
So, check:
$$\frac{x^{2} + 1}{x^{2}} = \frac{x^{2} + 1}{x^{2}}$$
- Yes
$$\frac{x^{2} + 1}{x^{2}} = - \frac{x^{2} + 1}{x^{2}}$$
- No
so, the function
is
even
The graph