Mister Exam
Graphing y = |x+2/x-3|
The solution
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| 2 |
f(x) = |x + - - 3|
| x |
$$f{\left(x \right)} = \left|{\left(x + \frac{2}{x}\right) - 3}\right|$$
f = |x + 2/x - 3|
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:
$$\left|{\left(x + \frac{2}{x}\right) - 3}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 2$$
so we need to solve the equation:
$$\left|{\left(x + \frac{2}{x}\right) - 3}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 2$$
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
$$\lim_{x \to -\infty} \left|{\left(x + \frac{2}{x}\right) - 3}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \left|{\left(x + \frac{2}{x}\right) - 3}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \left|{\left(x + \frac{2}{x}\right) - 3}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \left|{\left(x + \frac{2}{x}\right) - 3}\right| = \infty$$
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 + 2/x - 3|, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left|{\left(x + \frac{2}{x}\right) - 3}\right|}{x}\right) = -1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - x$$
$$\lim_{x \to \infty}\left(\frac{\left|{\left(x + \frac{2}{x}\right) - 3}\right|}{x}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x$$
$$\lim_{x \to -\infty}\left(\frac{\left|{\left(x + \frac{2}{x}\right) - 3}\right|}{x}\right) = -1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - x$$
$$\lim_{x \to \infty}\left(\frac{\left|{\left(x + \frac{2}{x}\right) - 3}\right|}{x}\right) = 1$$
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:
$$\left|{\left(x + \frac{2}{x}\right) - 3}\right| = \left|{x + 3 + \frac{2}{x}}\right|$$
- No
$$\left|{\left(x + \frac{2}{x}\right) - 3}\right| = - \left|{x + 3 + \frac{2}{x}}\right|$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left|{\left(x + \frac{2}{x}\right) - 3}\right| = \left|{x + 3 + \frac{2}{x}}\right|$$
- No
$$\left|{\left(x + \frac{2}{x}\right) - 3}\right| = - \left|{x + 3 + \frac{2}{x}}\right|$$
- No
so, the function
not is
neither even, nor odd