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  • How to use it?

  • Graphing y =:
  • x^3-12x+1
  • (x^2-x-2)/(x-2)
  • -x/2
  • (x^2-6x+13)/(x-3)

  • Identical expressions

  • two *(one / three)^x- one
  • 2 multiply by (1 divide by 3) to the power of x minus 1
  • two multiply by (one divide by three) to the power of x minus one
  • 2*(1/3)x-1
  • 2*1/3x-1
  • 2(1/3)^x-1
  • 2(1/3)x-1
  • 21/3x-1
  • 21/3^x-1
  • 2*(1 divide by 3)^x-1

  • Similar expressions

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

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

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          -x    
f(x) = 2*3   - 1
f(x)=−1+2(13)xf{\left(x \right)} = -1 + 2 \left(\frac{1}{3}\right)^{x} 
f = -1 + 2*(1/3)^x
The graph of the function
02468-8-6-4-2-1010200000-100000
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:
−1+2(13)x=0-1 + 2 \left(\frac{1}{3}\right)^{x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=log⁡(2)log⁡(3)x_{1} = \frac{\log{\left(2 \right)}}{\log{\left(3 \right)}}
Numerical solution
x1=0.630929753571457x_{1} = 0.630929753571457
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2*(1/3)^x - 1.
−1+2(13)0-1 + 2 \left(\frac{1}{3}\right)^{0}
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
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
−2⋅3−xlog⁡(3)=0- 2 \cdot 3^{- x} \log{\left(3 \right)} = 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
2⋅3−xlog⁡(3)2=02 \cdot 3^{- x} \log{\left(3 \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→−∞(−1+2(13)x)=∞\lim_{x \to -\infty}\left(-1 + 2 \left(\frac{1}{3}\right)^{x}\right) = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(−1+2(13)x)=−1\lim_{x \to \infty}\left(-1 + 2 \left(\frac{1}{3}\right)^{x}\right) = -1
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=−1y = -1
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2*(1/3)^x - 1, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(−1+2(13)xx)=−∞\lim_{x \to -\infty}\left(\frac{-1 + 2 \left(\frac{1}{3}\right)^{x}}{x}\right) = -\infty
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
lim⁡x→∞(−1+2(13)xx)=0\lim_{x \to \infty}\left(\frac{-1 + 2 \left(\frac{1}{3}\right)^{x}}{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:
−1+2(13)x=2⋅3x−1-1 + 2 \left(\frac{1}{3}\right)^{x} = 2 \cdot 3^{x} - 1
- No
−1+2(13)x=1−2⋅3x-1 + 2 \left(\frac{1}{3}\right)^{x} = 1 - 2 \cdot 3^{x}
- No
so, the function
not is
neither even, nor odd
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