Mister Exam
Graphing y = x-3/(x-4)(2x+1)
The solution
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3
f(x) = x - -----*(2*x + 1)
x - 4
$$f{\left(x \right)} = x - \frac{3}{x - 4} \left(2 x + 1\right)$$
f = x - 3/(x - 4)*(2*x + 1)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 4$$
$$x_{1} = 4$$
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:
$$x - \frac{3}{x - 4} \left(2 x + 1\right) = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 5 - 2 \sqrt{7}$$
$$x_{2} = 5 + 2 \sqrt{7}$$
Numerical solution
$$x_{1} = 10.2915026221292$$
$$x_{2} = -0.291502622129181$$
so we need to solve the equation:
$$x - \frac{3}{x - 4} \left(2 x + 1\right) = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 5 - 2 \sqrt{7}$$
$$x_{2} = 5 + 2 \sqrt{7}$$
Numerical solution
$$x_{1} = 10.2915026221292$$
$$x_{2} = -0.291502622129181$$
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
$$1 - \frac{6}{x - 4} + \frac{3 \left(2 x + 1\right)}{\left(x - 4\right)^{2}} = 0$$
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
$$1 - \frac{6}{x - 4} + \frac{3 \left(2 x + 1\right)}{\left(x - 4\right)^{2}} = 0$$
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
$$\frac{6 \left(2 - \frac{2 x + 1}{x - 4}\right)}{\left(x - 4\right)^{2}} = 0$$
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
$$\frac{6 \left(2 - \frac{2 x + 1}{x - 4}\right)}{\left(x - 4\right)^{2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 4$$
$$x_{1} = 4$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(x - \frac{3}{x - 4} \left(2 x + 1\right)\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x - \frac{3}{x - 4} \left(2 x + 1\right)\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(x - \frac{3}{x - 4} \left(2 x + 1\right)\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x - \frac{3}{x - 4} \left(2 x + 1\right)\right) = \infty$$
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 x - 3/(x - 4)*(2*x + 1), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{x - \frac{3}{x - 4} \left(2 x + 1\right)}{x}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x$$
$$\lim_{x \to \infty}\left(\frac{x - \frac{3}{x - 4} \left(2 x + 1\right)}{x}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x$$
$$\lim_{x \to -\infty}\left(\frac{x - \frac{3}{x - 4} \left(2 x + 1\right)}{x}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x$$
$$\lim_{x \to \infty}\left(\frac{x - \frac{3}{x - 4} \left(2 x + 1\right)}{x}\right) = 1$$
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:
$$x - \frac{3}{x - 4} \left(2 x + 1\right) = - x - \frac{3 \left(1 - 2 x\right)}{- x - 4}$$
- No
$$x - \frac{3}{x - 4} \left(2 x + 1\right) = x + \frac{3 \left(1 - 2 x\right)}{- x - 4}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$x - \frac{3}{x - 4} \left(2 x + 1\right) = - x - \frac{3 \left(1 - 2 x\right)}{- x - 4}$$
- No
$$x - \frac{3}{x - 4} \left(2 x + 1\right) = x + \frac{3 \left(1 - 2 x\right)}{- x - 4}$$
- No
so, the function
not is
neither even, nor odd