Mister Exam
Graphing y = sqrt(4-x)-sqrt(x+2)
The solution
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_______ _______ f(x) = \/ 4 - x - \/ x + 2
$$f{\left(x \right)} = \sqrt{4 - x} - \sqrt{x + 2}$$
f = sqrt(4 - x) - sqrt(x + 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:
$$\sqrt{4 - x} - \sqrt{x + 2} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
so we need to solve the equation:
$$\sqrt{4 - x} - \sqrt{x + 2} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
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
$$- \frac{1}{2 \sqrt{x + 2}} - \frac{1}{2 \sqrt{4 - x}} = 0$$
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
$$- \frac{1}{2 \sqrt{x + 2}} - \frac{1}{2 \sqrt{4 - x}} = 0$$
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
$$\frac{\frac{1}{\left(x + 2\right)^{\frac{3}{2}}} - \frac{1}{\left(4 - x\right)^{\frac{3}{2}}}}{4} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 1$$
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left(-\infty, 1\right]$$
Convex at the intervals
$$\left[1, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\frac{1}{\left(x + 2\right)^{\frac{3}{2}}} - \frac{1}{\left(4 - x\right)^{\frac{3}{2}}}}{4} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 1$$
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left(-\infty, 1\right]$$
Convex at the intervals
$$\left[1, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\sqrt{4 - x} - \sqrt{x + 2}\right) = - \infty \operatorname{sign}{\left(-1 + i \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = - \infty \operatorname{sign}{\left(-1 + i \right)}$$
$$\lim_{x \to \infty}\left(\sqrt{4 - x} - \sqrt{x + 2}\right) = \infty \operatorname{sign}{\left(-1 + i \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \infty \operatorname{sign}{\left(-1 + i \right)}$$
$$\lim_{x \to -\infty}\left(\sqrt{4 - x} - \sqrt{x + 2}\right) = - \infty \operatorname{sign}{\left(-1 + i \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = - \infty \operatorname{sign}{\left(-1 + i \right)}$$
$$\lim_{x \to \infty}\left(\sqrt{4 - x} - \sqrt{x + 2}\right) = \infty \operatorname{sign}{\left(-1 + i \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \infty \operatorname{sign}{\left(-1 + i \right)}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sqrt(4 - x) - sqrt(x + 2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt{4 - x} - \sqrt{x + 2}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{4 - x} - \sqrt{x + 2}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sqrt{4 - x} - \sqrt{x + 2}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{4 - x} - \sqrt{x + 2}}{x}\right) = 0$$
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:
$$\sqrt{4 - x} - \sqrt{x + 2} = - \sqrt{2 - x} + \sqrt{x + 4}$$
- No
$$\sqrt{4 - x} - \sqrt{x + 2} = \sqrt{2 - x} - \sqrt{x + 4}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sqrt{4 - x} - \sqrt{x + 2} = - \sqrt{2 - x} + \sqrt{x + 4}$$
- No
$$\sqrt{4 - x} - \sqrt{x + 2} = \sqrt{2 - x} - \sqrt{x + 4}$$
- No
so, the function
not is
neither even, nor odd