Mister Exam

Graphing y = log0,5(x-1)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = log(0.0)*5*(x - 1)
$$f{\left(x \right)} = 5 \log{\left(0.0 \right)} \left(x - 1\right)$$
f = (5*log(0.0))*(x - 1)
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:
$$5 \log{\left(0 \right)} \left(x - 1\right) = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (log(0)*5)*(x - 1).
$$\left(-1\right) 5 \log{\left(0 \right)}$$
The result:
$$f{\left(0 \right)} = \tilde{\infty}$$
sof doesn't intersect Y
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
$$5 \log{\left(0 \right)} = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\text{NaN} = 0$$
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
False

Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
False

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(0)*5)*(x - 1), divided by x at x->+oo and x ->-oo
False

Let's take the limit
so,
inclined asymptote on the left doesn’t exist
False

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:
$$5 \log{\left(0 \right)} \left(x - 1\right) = 5 \left(- x - 1\right) \log{\left(0 \right)}$$
- No
$$5 \log{\left(0 \right)} \left(x - 1\right) = - 5 \left(- x - 1\right) \log{\left(0 \right)}$$
- No
so, the function
not is
neither even, nor odd