Mister Exam
Graphing y = y=x22
The solution
The graph of the function
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X22 at f(x22) = 0
so we need to solve the equation:
$$x_{22} = 0$$
Solve this equation
The points of intersection with the axis X22:
Analytical solution
$$x_{221} = 0$$
Numerical solution
$$x_{221} = 0$$
so we need to solve the equation:
$$x_{22} = 0$$
Solve this equation
The points of intersection with the axis X22:
Analytical solution
$$x_{221} = 0$$
Numerical solution
$$x_{221} = 0$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x_{22}} f{\left(x_{22} \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x_{22}} f{\left(x_{22} \right)} = $$
the first derivative
$$1 = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x_{22}} f{\left(x_{22} \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x_{22}} f{\left(x_{22} \right)} = $$
the first derivative
$$1 = 0$$
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
$$\frac{d^{2}}{d x_{22}^{2}} f{\left(x_{22} \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x_{22}^{2}} f{\left(x_{22} \right)} = $$
the second derivative
$$0 = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x_{22}^{2}} f{\left(x_{22} \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x_{22}^{2}} f{\left(x_{22} \right)} = $$
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 x22->+oo and x22->-oo
$$\lim_{x_{22} \to -\infty} x_{22} = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x_{22} \to \infty} x_{22} = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x_{22} \to -\infty} x_{22} = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x_{22} \to \infty} x_{22} = \infty$$
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 x22, divided by x22 at x22->+oo and x22 ->-oo
$$\lim_{x_{22} \to -\infty} 1 = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x_{22}$$
$$\lim_{x_{22} \to \infty} 1 = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x_{22}$$
$$\lim_{x_{22} \to -\infty} 1 = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x_{22}$$
$$\lim_{x_{22} \to \infty} 1 = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x_{22}$$
Even and odd functions
Let's check, whether the function even or odd by using relations f(x22) = f(-x22) и f(x22) = -f(-x22).
So, check:
$$x_{22} = - x_{22}$$
- No
$$x_{22} = x_{22}$$
- Yes
so, the function
is
odd
So, check:
$$x_{22} = - x_{22}$$
- No
$$x_{22} = x_{22}$$
- Yes
so, the function
is
odd