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  • How to use it?

  • Graphing y =:
  • x/(x^2-4)
  • x-cbrt(x^2)
  • x^5/5-4x^3/3
  • (x-4)/(x+5)

  • Identical expressions

  • four *x/(((nine *x- one)^ two)^(one / three)-(nine *x- one)^(one / three)+ one)
  • 4 multiply by x divide by (((9 multiply by x minus 1) squared ) to the power of (1 divide by 3) minus (9 multiply by x minus 1) to the power of (1 divide by 3) plus 1)
  • four multiply by x divide by (((nine multiply by x minus one) to the power of two) to the power of (one divide by three) minus (nine multiply by x minus one) to the power of (one divide by three) plus one)
  • 4*x/(((9*x-1)2)(1/3)-(9*x-1)(1/3)+1)
  • 4*x/9*x-121/3-9*x-11/3+1
  • 4*x/(((9*x-1)²)^(1/3)-(9*x-1)^(1/3)+1)
  • 4*x/(((9*x-1) to the power of 2) to the power of (1/3)-(9*x-1) to the power of (1/3)+1)
  • 4x/(((9x-1)^2)^(1/3)-(9x-1)^(1/3)+1)
  • 4x/(((9x-1)2)(1/3)-(9x-1)(1/3)+1)
  • 4x/9x-121/3-9x-11/3+1
  • 4x/9x-1^2^1/3-9x-1^1/3+1
  • 4*x divide by (((9*x-1)^2)^(1 divide by 3)-(9*x-1)^(1 divide by 3)+1)

  • Similar expressions

  • 4*x/(((9*x-1)^2)^(1/3)-(9*x+1)^(1/3)+1)
  • 4*x/(((9*x-1)^2)^(1/3)-(9*x-1)^(1/3)-1)
  • 4*x/(((9*x+1)^2)^(1/3)-(9*x-1)^(1/3)+1)
  • 4*x/(((9*x-1)^2)^(1/3)+(9*x-1)^(1/3)+1)
Plotting a function graph/ 4*x/(((9*x-1)^2)^(1/3)-(9*x-1)^(1/3)+1)

Graphing y = 4*x/(((9*x-1)^2)^(1/3)-(9*x-1)^(1/3)+1)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
                      4*x               
f(x) = ---------------------------------
          ____________                  
       3 /          2    3 _________    
       \/  (9*x - 1)   - \/ 9*x - 1  + 1
$$f{\left(x \right)} = \frac{4 x}{\left(- \sqrt[3]{9 x - 1} + \sqrt[3]{\left(9 x - 1\right)^{2}}\right) + 1}$$ 
f = (4*x)/(-(9*x - 1)^(1/3) + ((9*x - 1)^2)^(1/3) + 1)
The graph of the function
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:
$$\frac{4 x}{\left(- \sqrt[3]{9 x - 1} + \sqrt[3]{\left(9 x - 1\right)^{2}}\right) + 1} = 0$$
Solve this equation
The points of intersection with the axis X:

Numerical solution
$$x_{1} = 0$$
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (4*x)/(((9*x - 1)^2)^(1/3) - (9*x - 1)^(1/3) + 1).
$$\frac{0 \cdot 4}{1 + \left(\sqrt[3]{\left(-1 + 0 \cdot 9\right)^{2}} - \sqrt[3]{-1 + 0 \cdot 9}\right)}$$
The result:
$$f{\left(0 \right)} = 0$$
The point:
(0, 0)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{4 x}{\left(- \sqrt[3]{9 x - 1} + \sqrt[3]{\left(9 x - 1\right)^{2}}\right) + 1}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{4 x}{\left(- \sqrt[3]{9 x - 1} + \sqrt[3]{\left(9 x - 1\right)^{2}}\right) + 1}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (4*x)/(((9*x - 1)^2)^(1/3) - (9*x - 1)^(1/3) + 1), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{4}{\left(- \sqrt[3]{9 x - 1} + \sqrt[3]{\left(9 x - 1\right)^{2}}\right) + 1}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{4}{\left(- \sqrt[3]{9 x - 1} + \sqrt[3]{\left(9 x - 1\right)^{2}}\right) + 1}\right) = 0$$
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:
$$\frac{4 x}{\left(- \sqrt[3]{9 x - 1} + \sqrt[3]{\left(9 x - 1\right)^{2}}\right) + 1} = - \frac{4 x}{- \sqrt[3]{- 9 x - 1} + \left|{9 x + 1}\right|^{\frac{2}{3}} + 1}$$
- No
$$\frac{4 x}{\left(- \sqrt[3]{9 x - 1} + \sqrt[3]{\left(9 x - 1\right)^{2}}\right) + 1} = \frac{4 x}{- \sqrt[3]{- 9 x - 1} + \left|{9 x + 1}\right|^{\frac{2}{3}} + 1}$$
- No
so, the function
not is
neither even, nor odd
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