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((1/3)^x)+2

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  • Graphing y =:
  • x^2+6x+10
  • x^2+4x+5
  • -x^2+2x-3
  • 16x^3+12x^2-5

  • Identical expressions

  • ((one / three)^x)+ two
  • ((1 divide by 3) to the power of x) plus 2
  • ((one divide by three) to the power of x) plus two
  • ((1/3)x)+2
  • 1/3x+2
  • 1/3^x+2
  • ((1 divide by 3)^x)+2

  • Similar expressions

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

Graphing y = ((1/3)^x)+2

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
        -x    
f(x) = 3   + 2
f(x)=2+(13)xf{\left(x \right)} = 2 + \left(\frac{1}{3}\right)^{x} 
f = 2 + (1/3)^x
The graph of the function
02468-8-6-4-2-10100100000
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:
2+(13)x=02 + \left(\frac{1}{3}\right)^{x} = 0
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (1/3)^x + 2.
(13)0+2\left(\frac{1}{3}\right)^{0} + 2
The result:
f(0)=3f{\left(0 \right)} = 3
The point:
(0, 3)
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
−3−xlog⁡(3)=0- 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
3−xlog⁡(3)2=03^{- 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→−∞(2+(13)x)=∞\lim_{x \to -\infty}\left(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→∞(2+(13)x)=2\lim_{x \to \infty}\left(2 + \left(\frac{1}{3}\right)^{x}\right) = 2
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=2y = 2
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (1/3)^x + 2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2+(13)xx)=−∞\lim_{x \to -\infty}\left(\frac{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→∞(2+(13)xx)=0\lim_{x \to \infty}\left(\frac{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:
2+(13)x=(13)−x+22 + \left(\frac{1}{3}\right)^{x} = \left(\frac{1}{3}\right)^{- x} + 2
- No
2+(13)x=−(13)−x−22 + \left(\frac{1}{3}\right)^{x} = - \left(\frac{1}{3}\right)^{- x} - 2
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = ((1/3)^x)+2
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