Mister Exam
Graphing y = 3*(log(3)(1-x/2))+1
The solution
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[src]
/ x\
f(x) = 3*log(3)*|1 - -| + 1
\ 2/
$$f{\left(x \right)} = 3 \left(- \frac{x}{2} + 1\right) \log{\left(3 \right)} + 1$$
f = 3*((-x/2 + 1)*log(3)) + 1
The graph of the function
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:
$$3 \left(- \frac{x}{2} + 1\right) \log{\left(3 \right)} + 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{2}{3 \log{\left(3 \right)}} + 2$$
Numerical solution
$$x_{1} = 2.60682615108456$$
so we need to solve the equation:
$$3 \left(- \frac{x}{2} + 1\right) \log{\left(3 \right)} + 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{2}{3 \log{\left(3 \right)}} + 2$$
Numerical solution
$$x_{1} = 2.60682615108456$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$- \frac{3 \log{\left(3 \right)}}{2} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$- \frac{3 \log{\left(3 \right)}}{2} = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$0 = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$0 = 0$$
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
$$\lim_{x \to -\infty}\left(3 \left(- \frac{x}{2} + 1\right) \log{\left(3 \right)} + 1\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(3 \left(- \frac{x}{2} + 1\right) \log{\left(3 \right)} + 1\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(3 \left(- \frac{x}{2} + 1\right) \log{\left(3 \right)} + 1\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(3 \left(- \frac{x}{2} + 1\right) \log{\left(3 \right)} + 1\right) = -\infty$$
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 3*(log(3)*(1 - x/2)) + 1, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{3 \left(- \frac{x}{2} + 1\right) \log{\left(3 \right)} + 1}{x}\right) = - \frac{3 \log{\left(3 \right)}}{2}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - \frac{3 x \log{\left(3 \right)}}{2}$$
$$\lim_{x \to \infty}\left(\frac{3 \left(- \frac{x}{2} + 1\right) \log{\left(3 \right)} + 1}{x}\right) = - \frac{3 \log{\left(3 \right)}}{2}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - \frac{3 x \log{\left(3 \right)}}{2}$$
$$\lim_{x \to -\infty}\left(\frac{3 \left(- \frac{x}{2} + 1\right) \log{\left(3 \right)} + 1}{x}\right) = - \frac{3 \log{\left(3 \right)}}{2}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - \frac{3 x \log{\left(3 \right)}}{2}$$
$$\lim_{x \to \infty}\left(\frac{3 \left(- \frac{x}{2} + 1\right) \log{\left(3 \right)} + 1}{x}\right) = - \frac{3 \log{\left(3 \right)}}{2}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - \frac{3 x \log{\left(3 \right)}}{2}$$
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:
$$3 \left(- \frac{x}{2} + 1\right) \log{\left(3 \right)} + 1 = 3 \left(\frac{x}{2} + 1\right) \log{\left(3 \right)} + 1$$
- No
$$3 \left(- \frac{x}{2} + 1\right) \log{\left(3 \right)} + 1 = - 3 \left(\frac{x}{2} + 1\right) \log{\left(3 \right)} - 1$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$3 \left(- \frac{x}{2} + 1\right) \log{\left(3 \right)} + 1 = 3 \left(\frac{x}{2} + 1\right) \log{\left(3 \right)} + 1$$
- No
$$3 \left(- \frac{x}{2} + 1\right) \log{\left(3 \right)} + 1 = - 3 \left(\frac{x}{2} + 1\right) \log{\left(3 \right)} - 1$$
- No
so, the function
not is
neither even, nor odd