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

Graphing y = -2+log((x-3)^2)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = -2 + log\(x - 3) /
f(x)=log⁡((x−3)2)−2f{\left(x \right)} = \log{\left(\left(x - 3\right)^{2} \right)} - 2 
f = log((x - 3)^2) - 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⁡((x−3)2)−2=0\log{\left(\left(x - 3\right)^{2} \right)} - 2 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=3−ex_{1} = 3 - e
x2=e+3x_{2} = e + 3
Numerical solution
x1=0.281718171540955x_{1} = 0.281718171540955
x2=5.71828182845905x_{2} = 5.71828182845905
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to -2 + log((x - 3)^2).
−2+log⁡((−3)2)-2 + \log{\left(\left(-3\right)^{2} \right)}
The result:
f(0)=−2+log⁡(9)f{\left(0 \right)} = -2 + \log{\left(9 \right)}
The point:
(0, -2 + log(9))
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
2x−6(x−3)2=0\frac{2 x - 6}{\left(x - 3\right)^{2}} = 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(x−3)2=0- \frac{2}{\left(x - 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→−∞(log⁡((x−3)2)−2)=∞\lim_{x \to -\infty}\left(\log{\left(\left(x - 3\right)^{2} \right)} - 2\right) = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(log⁡((x−3)2)−2)=∞\lim_{x \to \infty}\left(\log{\left(\left(x - 3\right)^{2} \right)} - 2\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 -2 + log((x - 3)^2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(log⁡((x−3)2)−2x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(\left(x - 3\right)^{2} \right)} - 2}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(log⁡((x−3)2)−2x)=0\lim_{x \to \infty}\left(\frac{\log{\left(\left(x - 3\right)^{2} \right)} - 2}{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⁡((x−3)2)−2=log⁡((−x−3)2)−2\log{\left(\left(x - 3\right)^{2} \right)} - 2 = \log{\left(\left(- x - 3\right)^{2} \right)} - 2
- No
log⁡((x−3)2)−2=2−log⁡((−x−3)2)\log{\left(\left(x - 3\right)^{2} \right)} - 2 = 2 - \log{\left(\left(- x - 3\right)^{2} \right)}
- No
so, the function
not is
neither even, nor odd
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