Mister Exam
Graphing y = log1/5(x2–3)
The solution
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log(1)
f(x2) = ------*(x2 - 3)
5
$$f{\left(x_{2} \right)} = \frac{\log{\left(1 \right)}}{5} \left(x_{2} - 3\right)$$
f(x2) = (log(1)/5)*(x2 - 3)
The graph of the function
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X2 at f(x2) = 0
so we need to solve the equation:
$$\frac{\log{\left(1 \right)}}{5} \left(x_{2} - 3\right) = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X2
so we need to solve the equation:
$$\frac{\log{\left(1 \right)}}{5} \left(x_{2} - 3\right) = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X2
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x_{2}} f{\left(x_{2} \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x_{2}} f{\left(x_{2} \right)} = $$
the first derivative
$$\frac{\log{\left(1 \right)}}{5} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x_{2}} f{\left(x_{2} \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x_{2}} f{\left(x_{2} \right)} = $$
the first derivative
$$\frac{\log{\left(1 \right)}}{5} = 0$$
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
$$\frac{d^{2}}{d x_{2}^{2}} f{\left(x_{2} \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x_{2}^{2}} f{\left(x_{2} \right)} = $$
the second derivative
$$0 = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x_{2}^{2}} f{\left(x_{2} \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x_{2}^{2}} f{\left(x_{2} \right)} = $$
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 x2->+oo and x2->-oo
$$\lim_{x_{2} \to -\infty}\left(\frac{\log{\left(1 \right)}}{5} \left(x_{2} - 3\right)\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x_{2} \to \infty}\left(\frac{\log{\left(1 \right)}}{5} \left(x_{2} - 3\right)\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x_{2} \to -\infty}\left(\frac{\log{\left(1 \right)}}{5} \left(x_{2} - 3\right)\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x_{2} \to \infty}\left(\frac{\log{\left(1 \right)}}{5} \left(x_{2} - 3\right)\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (log(1)/5)*(x2 - 3), divided by x2 at x2->+oo and x2 ->-oo
$$\lim_{x_{2} \to -\infty}\left(\frac{\left(x_{2} - 3\right) \log{\left(1 \right)}}{5 x_{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x_{2} \to \infty}\left(\frac{\left(x_{2} - 3\right) \log{\left(1 \right)}}{5 x_{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x_{2} \to -\infty}\left(\frac{\left(x_{2} - 3\right) \log{\left(1 \right)}}{5 x_{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x_{2} \to \infty}\left(\frac{\left(x_{2} - 3\right) \log{\left(1 \right)}}{5 x_{2}}\right) = 0$$
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(x2) = f(-x2) и f(x2) = -f(-x2).
So, check:
$$\frac{\log{\left(1 \right)}}{5} \left(x_{2} - 3\right) = \frac{\left(- x_{2} - 3\right) \log{\left(1 \right)}}{5}$$
- No
$$\frac{\log{\left(1 \right)}}{5} \left(x_{2} - 3\right) = - \frac{\left(- x_{2} - 3\right) \log{\left(1 \right)}}{5}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\log{\left(1 \right)}}{5} \left(x_{2} - 3\right) = \frac{\left(- x_{2} - 3\right) \log{\left(1 \right)}}{5}$$
- No
$$\frac{\log{\left(1 \right)}}{5} \left(x_{2} - 3\right) = - \frac{\left(- x_{2} - 3\right) \log{\left(1 \right)}}{5}$$
- No
so, the function
not is
neither even, nor odd