Mister Exam
Graphing y = log(2,(x+3))-5
The solution
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f(x) = log(2, x + 3) - 5
$$f{\left(x \right)} = \log{\left(2 \right)} - 5$$
Eq(f, log(2, x + 3) - 5)
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:
$$\log{\left(2 \right)} - 5 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -3 + \sqrt[5]{2}$$
Numerical solution
$$x_{1} = -1.85130164500296$$
so we need to solve the equation:
$$\log{\left(2 \right)} - 5 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -3 + \sqrt[5]{2}$$
Numerical solution
$$x_{1} = -1.85130164500296$$
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
$$\left. \frac{d}{d \xi_{2}} \frac{\log{\left(2 \right)}}{\log{\left(\xi_{2} \right)}} \right|_{\substack{ \xi_{2}=x + 3 }} = 0$$
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
$$\left. \frac{d}{d \xi_{2}} \frac{\log{\left(2 \right)}}{\log{\left(\xi_{2} \right)}} \right|_{\substack{ \xi_{2}=x + 3 }} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\log{\left(2 \right)} - 5\right) = -5$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -5$$
$$\lim_{x \to \infty}\left(\log{\left(2 \right)} - 5\right) = -5$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = -5$$
$$\lim_{x \to -\infty}\left(\log{\left(2 \right)} - 5\right) = -5$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -5$$
$$\lim_{x \to \infty}\left(\log{\left(2 \right)} - 5\right) = -5$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = -5$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(2, x + 3) - 5, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(2 \right)} - 5}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(2 \right)} - 5}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\log{\left(2 \right)} - 5}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(2 \right)} - 5}{x}\right) = 0$$
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:
$$\log{\left(2 \right)} - 5 = -5 + \frac{\log{\left(2 \right)}}{\log{\left(3 - x \right)}}$$
- No
$$\log{\left(2 \right)} - 5 = 5 - \frac{\log{\left(2 \right)}}{\log{\left(3 - x \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\log{\left(2 \right)} - 5 = -5 + \frac{\log{\left(2 \right)}}{\log{\left(3 - x \right)}}$$
- No
$$\log{\left(2 \right)} - 5 = 5 - \frac{\log{\left(2 \right)}}{\log{\left(3 - x \right)}}$$
- No
so, the function
not is
neither even, nor odd