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  • How to use it?

  • Graphing y =:
  • x+|x|
  • x/((x-1)*(x-4))
  • x(x-1)^3
  • x*sqrt(2-x)

  • Identical expressions

  • log(3x- one)*sqrt(two)
  • logarithm of (3x minus 1) multiply by square root of (2)
  • logarithm of (3x minus one) multiply by square root of (two)
  • log(3x-1)*√(2)
  • log(3x-1)sqrt(2)
  • log3x-1sqrt2

  • Similar expressions

  • log(3x+1)*sqrt(2)
Plotting a function graph/ log(3x-1)*sqrt(2)

Graphing y = log(3x-1)*sqrt(2)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
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f(x) = log(3*x - 1)*\/ 2
f(x)=2log⁡(3x−1)f{\left(x \right)} = \sqrt{2} \log{\left(3 x - 1 \right)} 
f = sqrt(2)*log(3*x - 1)
The graph of the function
02468-8-6-4-2-1010-1010
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:
2log⁡(3x−1)=0\sqrt{2} \log{\left(3 x - 1 \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=23x_{1} = \frac{2}{3}
Numerical solution
x1=0.666666666666667x_{1} = 0.666666666666667
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(3*x - 1)*sqrt(2).
2log⁡(−1+0⋅3)\sqrt{2} \log{\left(-1 + 0 \cdot 3 \right)}
The result:
f(0)=2iπf{\left(0 \right)} = \sqrt{2} i \pi
The point:
(0, pi*i*sqrt(2))
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
323x−1=0\frac{3 \sqrt{2}}{3 x - 1} = 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
−92(3x−1)2=0- \frac{9 \sqrt{2}}{\left(3 x - 1\right)^{2}} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(2log⁡(3x−1))=∞\lim_{x \to -\infty}\left(\sqrt{2} \log{\left(3 x - 1 \right)}\right) = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(2log⁡(3x−1))=∞\lim_{x \to \infty}\left(\sqrt{2} \log{\left(3 x - 1 \right)}\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 log(3*x - 1)*sqrt(2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2log⁡(3x−1)x)=0\lim_{x \to -\infty}\left(\frac{\sqrt{2} \log{\left(3 x - 1 \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(2log⁡(3x−1)x)=0\lim_{x \to \infty}\left(\frac{\sqrt{2} \log{\left(3 x - 1 \right)}}{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:
2log⁡(3x−1)=2log⁡(−3x−1)\sqrt{2} \log{\left(3 x - 1 \right)} = \sqrt{2} \log{\left(- 3 x - 1 \right)}
- No
2log⁡(3x−1)=−2log⁡(−3x−1)\sqrt{2} \log{\left(3 x - 1 \right)} = - \sqrt{2} \log{\left(- 3 x - 1 \right)}
- No
so, the function
not is
neither even, nor odd
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