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Graphing y = log((3*x+4),2)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = log(3*x + 4, 2)
f(x)=log(3x+4)f{\left(x \right)} = \log{\left(3 x + 4 \right)}
Eq(f, log(3*x + 4, 2))
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:
log(3x+4)=0\log{\left(3 x + 4 \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=1x_{1} = -1
Numerical solution
x1=1x_{1} = -1
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 + 4, 2).
log(03+4)\log{\left(0 \cdot 3 + 4 \right)}
The result:
f(0)=2f{\left(0 \right)} = 2
The point:
(0, 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
33x+4=0\frac{3}{3 x + 4} = 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
9(3x+4)2=0- \frac{9}{\left(3 x + 4\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
limxlog(3x+4)=\lim_{x \to -\infty} \log{\left(3 x + 4 \right)} = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
limxlog(3x+4)=\lim_{x \to \infty} \log{\left(3 x + 4 \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 + 4, 2), divided by x at x->+oo and x ->-oo
limx(log(3x+4)x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(3 x + 4 \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
limx(log(3x+4)x)=0\lim_{x \to \infty}\left(\frac{\log{\left(3 x + 4 \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:
log(3x+4)=log(43x)log(2)\log{\left(3 x + 4 \right)} = \frac{\log{\left(4 - 3 x \right)}}{\log{\left(2 \right)}}
- No
log(3x+4)=log(43x)log(2)\log{\left(3 x + 4 \right)} = - \frac{\log{\left(4 - 3 x \right)}}{\log{\left(2 \right)}}
- No
so, the function
not is
neither even, nor odd