Mister Exam

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  • How to use it?

  • Graphing y =:
  • x^4-4*x^2
  • x^4-4x^2+2
  • x^4+2x^3
  • -x^4+2x^2
  • Identical expressions

  • four /sqrtx^ two + eight
  • 4 divide by square root of x squared plus 8
  • four divide by square root of x to the power of two plus eight
  • 4/√x^2+8
  • 4/sqrtx2+8
  • 4/sqrtx²+8
  • 4/sqrtx to the power of 2+8
  • 4 divide by sqrtx^2+8
  • Similar expressions

  • 4/sqrtx^2-8

Graphing y = 4/sqrtx^2+8

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src]
         4       
f(x) = ------ + 8
            2    
         ___     
       \/ x      
$$f{\left(x \right)} = 8 + \frac{4}{\left(\sqrt{x}\right)^{2}}$$
f = 8 + 4/(sqrt(x))^2
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
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:
$$8 + \frac{4}{\left(\sqrt{x}\right)^{2}} = 0$$
Solve this equation
The points of intersection with the axis X:

Analytical solution
$$x_{1} = - \frac{1}{2}$$
Numerical solution
$$x_{1} = -0.5$$
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 4/(sqrt(x))^2 + 8.
$$\frac{4}{\left(\sqrt{0}\right)^{2}} + 8$$
The result:
$$f{\left(0 \right)} = \tilde{\infty}$$
sof doesn't intersect Y
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- \frac{4}{x^{2}} = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{8}{x^{3}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(8 + \frac{4}{\left(\sqrt{x}\right)^{2}}\right) = 8$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 8$$
$$\lim_{x \to \infty}\left(8 + \frac{4}{\left(\sqrt{x}\right)^{2}}\right) = 8$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 8$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 4/(sqrt(x))^2 + 8, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{8 + \frac{4}{\left(\sqrt{x}\right)^{2}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{8 + \frac{4}{\left(\sqrt{x}\right)^{2}}}{x}\right) = 0$$
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:
$$8 + \frac{4}{\left(\sqrt{x}\right)^{2}} = 8 - \frac{4}{x}$$
- No
$$8 + \frac{4}{\left(\sqrt{x}\right)^{2}} = -8 + \frac{4}{x}$$
- No
so, the function
not is
neither even, nor odd