Mister Exam
Graphing y = (17x+20)/(x^2+8x-9)
The solution
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17*x + 20
f(x) = ------------
2
x + 8*x - 9
$$f{\left(x \right)} = \frac{17 x + 20}{\left(x^{2} + 8 x\right) - 9}$$
f = (17*x + 20)/(x^2 + 8*x - 9)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -9$$
$$x_{2} = 1$$
$$x_{1} = -9$$
$$x_{2} = 1$$
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:
$$\frac{17 x + 20}{\left(x^{2} + 8 x\right) - 9} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{20}{17}$$
Numerical solution
$$x_{1} = -1.17647058823529$$
so we need to solve the equation:
$$\frac{17 x + 20}{\left(x^{2} + 8 x\right) - 9} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{20}{17}$$
Numerical solution
$$x_{1} = -1.17647058823529$$
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
$$\frac{\left(- 2 x - 8\right) \left(17 x + 20\right)}{\left(\left(x^{2} + 8 x\right) - 9\right)^{2}} + \frac{17}{\left(x^{2} + 8 x\right) - 9} = 0$$
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
$$\frac{\left(- 2 x - 8\right) \left(17 x + 20\right)}{\left(\left(x^{2} + 8 x\right) - 9\right)^{2}} + \frac{17}{\left(x^{2} + 8 x\right) - 9} = 0$$
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
$$\frac{2 \left(- 34 x + \left(17 x + 20\right) \left(\frac{4 \left(x + 4\right)^{2}}{x^{2} + 8 x - 9} - 1\right) - 136\right)}{\left(x^{2} + 8 x - 9\right)^{2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\sqrt[3]{654493}}{17} - \frac{20}{17} + \frac{\sqrt[3]{182077}}{17}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -9$$
$$x_{2} = 1$$
$$\lim_{x \to -9^-}\left(\frac{2 \left(- 34 x + \left(17 x + 20\right) \left(\frac{4 \left(x + 4\right)^{2}}{x^{2} + 8 x - 9} - 1\right) - 136\right)}{\left(x^{2} + 8 x - 9\right)^{2}}\right) = -\infty$$
$$\lim_{x \to -9^+}\left(\frac{2 \left(- 34 x + \left(17 x + 20\right) \left(\frac{4 \left(x + 4\right)^{2}}{x^{2} + 8 x - 9} - 1\right) - 136\right)}{\left(x^{2} + 8 x - 9\right)^{2}}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = -9$$
- is an inflection point
$$\lim_{x \to 1^-}\left(\frac{2 \left(- 34 x + \left(17 x + 20\right) \left(\frac{4 \left(x + 4\right)^{2}}{x^{2} + 8 x - 9} - 1\right) - 136\right)}{\left(x^{2} + 8 x - 9\right)^{2}}\right) = -\infty$$
$$\lim_{x \to 1^+}\left(\frac{2 \left(- 34 x + \left(17 x + 20\right) \left(\frac{4 \left(x + 4\right)^{2}}{x^{2} + 8 x - 9} - 1\right) - 136\right)}{\left(x^{2} + 8 x - 9\right)^{2}}\right) = \infty$$
- the limits are not equal, so
$$x_{2} = 1$$
- is an inflection point
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
$$\left(-\infty, - \frac{\sqrt[3]{654493}}{17} - \frac{20}{17} + \frac{\sqrt[3]{182077}}{17}\right]$$
Convex at the intervals
$$\left[- \frac{\sqrt[3]{654493}}{17} - \frac{20}{17} + \frac{\sqrt[3]{182077}}{17}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{2 \left(- 34 x + \left(17 x + 20\right) \left(\frac{4 \left(x + 4\right)^{2}}{x^{2} + 8 x - 9} - 1\right) - 136\right)}{\left(x^{2} + 8 x - 9\right)^{2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\sqrt[3]{654493}}{17} - \frac{20}{17} + \frac{\sqrt[3]{182077}}{17}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -9$$
$$x_{2} = 1$$
$$\lim_{x \to -9^-}\left(\frac{2 \left(- 34 x + \left(17 x + 20\right) \left(\frac{4 \left(x + 4\right)^{2}}{x^{2} + 8 x - 9} - 1\right) - 136\right)}{\left(x^{2} + 8 x - 9\right)^{2}}\right) = -\infty$$
$$\lim_{x \to -9^+}\left(\frac{2 \left(- 34 x + \left(17 x + 20\right) \left(\frac{4 \left(x + 4\right)^{2}}{x^{2} + 8 x - 9} - 1\right) - 136\right)}{\left(x^{2} + 8 x - 9\right)^{2}}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = -9$$
- is an inflection point
$$\lim_{x \to 1^-}\left(\frac{2 \left(- 34 x + \left(17 x + 20\right) \left(\frac{4 \left(x + 4\right)^{2}}{x^{2} + 8 x - 9} - 1\right) - 136\right)}{\left(x^{2} + 8 x - 9\right)^{2}}\right) = -\infty$$
$$\lim_{x \to 1^+}\left(\frac{2 \left(- 34 x + \left(17 x + 20\right) \left(\frac{4 \left(x + 4\right)^{2}}{x^{2} + 8 x - 9} - 1\right) - 136\right)}{\left(x^{2} + 8 x - 9\right)^{2}}\right) = \infty$$
- the limits are not equal, so
$$x_{2} = 1$$
- is an inflection point
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
$$\left(-\infty, - \frac{\sqrt[3]{654493}}{17} - \frac{20}{17} + \frac{\sqrt[3]{182077}}{17}\right]$$
Convex at the intervals
$$\left[- \frac{\sqrt[3]{654493}}{17} - \frac{20}{17} + \frac{\sqrt[3]{182077}}{17}, \infty\right)$$
Vertical asymptotes
Have:
$$x_{1} = -9$$
$$x_{2} = 1$$
$$x_{1} = -9$$
$$x_{2} = 1$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{17 x + 20}{\left(x^{2} + 8 x\right) - 9}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{17 x + 20}{\left(x^{2} + 8 x\right) - 9}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{17 x + 20}{\left(x^{2} + 8 x\right) - 9}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{17 x + 20}{\left(x^{2} + 8 x\right) - 9}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (17*x + 20)/(x^2 + 8*x - 9), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{17 x + 20}{x \left(\left(x^{2} + 8 x\right) - 9\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{17 x + 20}{x \left(\left(x^{2} + 8 x\right) - 9\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{17 x + 20}{x \left(\left(x^{2} + 8 x\right) - 9\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{17 x + 20}{x \left(\left(x^{2} + 8 x\right) - 9\right)}\right) = 0$$
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:
$$\frac{17 x + 20}{\left(x^{2} + 8 x\right) - 9} = \frac{20 - 17 x}{x^{2} - 8 x - 9}$$
- No
$$\frac{17 x + 20}{\left(x^{2} + 8 x\right) - 9} = - \frac{20 - 17 x}{x^{2} - 8 x - 9}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{17 x + 20}{\left(x^{2} + 8 x\right) - 9} = \frac{20 - 17 x}{x^{2} - 8 x - 9}$$
- No
$$\frac{17 x + 20}{\left(x^{2} + 8 x\right) - 9} = - \frac{20 - 17 x}{x^{2} - 8 x - 9}$$
- No
so, the function
not is
neither even, nor odd