Mister Exam
Graphing y = y=ln(x2+3)
The solution
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f(x2) = log(x2 + 3)
$$f{\left(x_{2} \right)} = \log{\left(x_{2} + 3 \right)}$$
f(x2) = log(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:
$$\log{\left(x_{2} + 3 \right)} = 0$$
Solve this equation
The points of intersection with the axis X2:
Analytical solution
$$x_{21} = -2$$
Numerical solution
$$x_{21} = -2$$
so we need to solve the equation:
$$\log{\left(x_{2} + 3 \right)} = 0$$
Solve this equation
The points of intersection with the axis X2:
Analytical solution
$$x_{21} = -2$$
Numerical solution
$$x_{21} = -2$$
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
$$\frac{1}{x_{2} + 3} = 0$$
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
$$\frac{1}{x_{2} + 3} = 0$$
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
$$- \frac{1}{\left(x_{2} + 3\right)^{2}} = 0$$
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
$$- \frac{1}{\left(x_{2} + 3\right)^{2}} = 0$$
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 x2->+oo and x2->-oo
$$\lim_{x_{2} \to -\infty} \log{\left(x_{2} + 3 \right)} = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x_{2} \to \infty} \log{\left(x_{2} + 3 \right)} = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x_{2} \to -\infty} \log{\left(x_{2} + 3 \right)} = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x_{2} \to \infty} \log{\left(x_{2} + 3 \right)} = \infty$$
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 log(x2 + 3), divided by x2 at x2->+oo and x2 ->-oo
$$\lim_{x_{2} \to -\infty}\left(\frac{\log{\left(x_{2} + 3 \right)}}{x_{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x_{2} \to \infty}\left(\frac{\log{\left(x_{2} + 3 \right)}}{x_{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x_{2} \to -\infty}\left(\frac{\log{\left(x_{2} + 3 \right)}}{x_{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x_{2} \to \infty}\left(\frac{\log{\left(x_{2} + 3 \right)}}{x_{2}}\right) = 0$$
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:
$$\log{\left(x_{2} + 3 \right)} = \log{\left(3 - x_{2} \right)}$$
- No
$$\log{\left(x_{2} + 3 \right)} = - \log{\left(3 - x_{2} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\log{\left(x_{2} + 3 \right)} = \log{\left(3 - x_{2} \right)}$$
- No
$$\log{\left(x_{2} + 3 \right)} = - \log{\left(3 - x_{2} \right)}$$
- No
so, the function
not is
neither even, nor odd