Mister Exam
Graphing y = (1-x)*(x+2)/x-3
The solution
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(1 - x)*(x + 2)
f(x) = --------------- - 3
x
$$f{\left(x \right)} = -3 + \frac{\left(1 - x\right) \left(x + 2\right)}{x}$$
f = -3 + ((1 - x)*(x + 2))/x
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$-3 + \frac{\left(1 - x\right) \left(x + 2\right)}{x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -2 + \sqrt{6}$$
$$x_{2} = - \sqrt{6} - 2$$
Numerical solution
$$x_{1} = 0.449489742783178$$
$$x_{2} = -4.44948974278318$$
so we need to solve the equation:
$$-3 + \frac{\left(1 - x\right) \left(x + 2\right)}{x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -2 + \sqrt{6}$$
$$x_{2} = - \sqrt{6} - 2$$
Numerical solution
$$x_{1} = 0.449489742783178$$
$$x_{2} = -4.44948974278318$$
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
$$\frac{- 2 x - 1}{x} - \frac{\left(1 - x\right) \left(x + 2\right)}{x^{2}} = 0$$
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
$$\frac{- 2 x - 1}{x} - \frac{\left(1 - x\right) \left(x + 2\right)}{x^{2}} = 0$$
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
$$\frac{-2 + \frac{x - 1}{x} + \frac{x + 2}{x} + \frac{2 x + 1}{x} - \frac{2 \left(x - 1\right) \left(x + 2\right)}{x^{2}}}{x} = 0$$
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
$$\frac{-2 + \frac{x - 1}{x} + \frac{x + 2}{x} + \frac{2 x + 1}{x} - \frac{2 \left(x - 1\right) \left(x + 2\right)}{x^{2}}}{x} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(-3 + \frac{\left(1 - x\right) \left(x + 2\right)}{x}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(-3 + \frac{\left(1 - x\right) \left(x + 2\right)}{x}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(-3 + \frac{\left(1 - x\right) \left(x + 2\right)}{x}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(-3 + \frac{\left(1 - x\right) \left(x + 2\right)}{x}\right) = -\infty$$
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 ((1 - x)*(x + 2))/x - 3, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{-3 + \frac{\left(1 - x\right) \left(x + 2\right)}{x}}{x}\right) = -1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - x$$
$$\lim_{x \to \infty}\left(\frac{-3 + \frac{\left(1 - x\right) \left(x + 2\right)}{x}}{x}\right) = -1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - x$$
$$\lim_{x \to -\infty}\left(\frac{-3 + \frac{\left(1 - x\right) \left(x + 2\right)}{x}}{x}\right) = -1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - x$$
$$\lim_{x \to \infty}\left(\frac{-3 + \frac{\left(1 - x\right) \left(x + 2\right)}{x}}{x}\right) = -1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - x$$
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:
$$-3 + \frac{\left(1 - x\right) \left(x + 2\right)}{x} = -3 - \frac{\left(2 - x\right) \left(x + 1\right)}{x}$$
- No
$$-3 + \frac{\left(1 - x\right) \left(x + 2\right)}{x} = 3 + \frac{\left(2 - x\right) \left(x + 1\right)}{x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$-3 + \frac{\left(1 - x\right) \left(x + 2\right)}{x} = -3 - \frac{\left(2 - x\right) \left(x + 1\right)}{x}$$
- No
$$-3 + \frac{\left(1 - x\right) \left(x + 2\right)}{x} = 3 + \frac{\left(2 - x\right) \left(x + 1\right)}{x}$$
- No
so, the function
not is
neither even, nor odd