Mister Exam
Graphing y = sqrt(10)+x-sqrt(2)-x
The solution
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____ ___ f(x) = \/ 10 + x - \/ 2 - x
$$f{\left(x \right)} = - x + \left(\left(x + \sqrt{10}\right) - \sqrt{2}\right)$$
f = -x + x + sqrt(10) - sqrt(2)
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:
$$- x + \left(\left(x + \sqrt{10}\right) - \sqrt{2}\right) = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$- x + \left(\left(x + \sqrt{10}\right) - \sqrt{2}\right) = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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
$$0 = 0$$
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
$$0 = 0$$
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
$$0 = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$0 = 0$$
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 x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(- x + \left(\left(x + \sqrt{10}\right) - \sqrt{2}\right)\right) = - \sqrt{2} + \sqrt{10}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = - \sqrt{2} + \sqrt{10}$$
$$\lim_{x \to \infty}\left(- x + \left(\left(x + \sqrt{10}\right) - \sqrt{2}\right)\right) = - \sqrt{2} + \sqrt{10}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = - \sqrt{2} + \sqrt{10}$$
$$\lim_{x \to -\infty}\left(- x + \left(\left(x + \sqrt{10}\right) - \sqrt{2}\right)\right) = - \sqrt{2} + \sqrt{10}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = - \sqrt{2} + \sqrt{10}$$
$$\lim_{x \to \infty}\left(- x + \left(\left(x + \sqrt{10}\right) - \sqrt{2}\right)\right) = - \sqrt{2} + \sqrt{10}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = - \sqrt{2} + \sqrt{10}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sqrt(10) + x - sqrt(2) - x, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{- x + \left(\left(x + \sqrt{10}\right) - \sqrt{2}\right)}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{- x + \left(\left(x + \sqrt{10}\right) - \sqrt{2}\right)}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{- x + \left(\left(x + \sqrt{10}\right) - \sqrt{2}\right)}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{- x + \left(\left(x + \sqrt{10}\right) - \sqrt{2}\right)}{x}\right) = 0$$
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:
$$- x + \left(\left(x + \sqrt{10}\right) - \sqrt{2}\right) = - \sqrt{2} + \sqrt{10}$$
- No
$$- x + \left(\left(x + \sqrt{10}\right) - \sqrt{2}\right) = - \sqrt{10} + \sqrt{2}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- x + \left(\left(x + \sqrt{10}\right) - \sqrt{2}\right) = - \sqrt{2} + \sqrt{10}$$
- No
$$- x + \left(\left(x + \sqrt{10}\right) - \sqrt{2}\right) = - \sqrt{10} + \sqrt{2}$$
- No
so, the function
not is
neither even, nor odd