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Plotting a function graph/ log(5)^(x+1)-1

Graphing y = log(5)^(x+1)-1

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          x + 1       
f(x) = log     (5) - 1
f(x)=log⁡(5)x+1−1f{\left(x \right)} = \log{\left(5 \right)}^{x + 1} - 1 
f = log(5)^(x + 1) - 1
The graph of the function
02468-8-6-4-2-1010-200200
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:
log⁡(5)x+1−1=0\log{\left(5 \right)}^{x + 1} - 1 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−1x_{1} = -1
Numerical solution
x1=−0.999999999998895x_{1} = -0.999999999998895
x2=−1x_{2} = -1
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(5)^(x + 1) - 1.
−1+log⁡(5)1-1 + \log{\left(5 \right)}^{1}
The result:
f(0)=−1+log⁡(5)f{\left(0 \right)} = -1 + \log{\left(5 \right)}
The point:
(0, -1 + log(5))
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
log⁡(5)x+1log⁡(log⁡(5))=0\log{\left(5 \right)}^{x + 1} \log{\left(\log{\left(5 \right)} \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
log⁡(5)x+1log⁡(log⁡(5))2=0\log{\left(5 \right)}^{x + 1} \log{\left(\log{\left(5 \right)} \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→−∞(log⁡(5)x+1−1)=−1\lim_{x \to -\infty}\left(\log{\left(5 \right)}^{x + 1} - 1\right) = -1
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=−1y = -1
lim⁡x→∞(log⁡(5)x+1−1)=∞\lim_{x \to \infty}\left(\log{\left(5 \right)}^{x + 1} - 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 log(5)^(x + 1) - 1, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(log⁡(5)x+1−1x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(5 \right)}^{x + 1} - 1}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(log⁡(5)x+1−1x)=∞\lim_{x \to \infty}\left(\frac{\log{\left(5 \right)}^{x + 1} - 1}{x}\right) = \infty
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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:
log⁡(5)x+1−1=log⁡(5)1−x−1\log{\left(5 \right)}^{x + 1} - 1 = \log{\left(5 \right)}^{1 - x} - 1
- No
log⁡(5)x+1−1=1−log⁡(5)1−x\log{\left(5 \right)}^{x + 1} - 1 = 1 - \log{\left(5 \right)}^{1 - x}
- No
so, the function
not is
neither even, nor odd
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