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  • Graphing y =:
  • x^4-2
  • (x-3)/x
  • x^3-x^2-2x
  • |x+3|-|x-1|+x+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
Plotting a function graph/ 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(x)=8+4(x)2f{\left(x \right)} = 8 + \frac{4}{\left(\sqrt{x}\right)^{2}} 
f = 8 + 4/(sqrt(x))^2
The graph of the function
02468-8-6-4-2-1010-200200
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:
8+4(x)2=08 + \frac{4}{\left(\sqrt{x}\right)^{2}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−12x_{1} = - \frac{1}{2}
Numerical solution
x1=−0.5x_{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.
4(0)2+8\frac{4}{\left(\sqrt{0}\right)^{2}} + 8
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
−4x2=0- \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
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
8x3=0\frac{8}{x^{3}} = 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→−∞(8+4(x)2)=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 left:
y=8y = 8
lim⁡x→∞(8+4(x)2)=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=8y = 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→−∞(8+4(x)2x)=0\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→∞(8+4(x)2x)=0\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+4(x)2=8−4x8 + \frac{4}{\left(\sqrt{x}\right)^{2}} = 8 - \frac{4}{x}
- No
8+4(x)2=−8+4x8 + \frac{4}{\left(\sqrt{x}\right)^{2}} = -8 + \frac{4}{x}
- No
so, the function
not is
neither even, nor odd
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