Mister Exam
Graphing y = y=sqrt(x^2-3x+2)+(2x)/(x^2+2x+3)
The solution
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/ 2 2*x
f(x) = \/ x - 3*x + 2 + ------------
2
x + 2*x + 3
$$f{\left(x \right)} = \frac{2 x}{\left(x^{2} + 2 x\right) + 3} + \sqrt{\left(x^{2} - 3 x\right) + 2}$$
f = (2*x)/(x^2 + 2*x + 3) + sqrt(x^2 - 3*x + 2)
The graph of the function
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{2 x}{\left(x^{2} + 2 x\right) + 3} + \sqrt{\left(x^{2} - 3 x\right) + 2}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{2 x}{\left(x^{2} + 2 x\right) + 3} + \sqrt{\left(x^{2} - 3 x\right) + 2}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{2 x}{\left(x^{2} + 2 x\right) + 3} + \sqrt{\left(x^{2} - 3 x\right) + 2}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{2 x}{\left(x^{2} + 2 x\right) + 3} + \sqrt{\left(x^{2} - 3 x\right) + 2}\right) = \infty$$
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 sqrt(x^2 - 3*x + 2) + (2*x)/(x^2 + 2*x + 3), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{2 x}{\left(x^{2} + 2 x\right) + 3} + \sqrt{\left(x^{2} - 3 x\right) + 2}}{x}\right) = -1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - x$$
$$\lim_{x \to \infty}\left(\frac{\frac{2 x}{\left(x^{2} + 2 x\right) + 3} + \sqrt{\left(x^{2} - 3 x\right) + 2}}{x}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x$$
$$\lim_{x \to -\infty}\left(\frac{\frac{2 x}{\left(x^{2} + 2 x\right) + 3} + \sqrt{\left(x^{2} - 3 x\right) + 2}}{x}\right) = -1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - x$$
$$\lim_{x \to \infty}\left(\frac{\frac{2 x}{\left(x^{2} + 2 x\right) + 3} + \sqrt{\left(x^{2} - 3 x\right) + 2}}{x}\right) = 1$$
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:
$$\frac{2 x}{\left(x^{2} + 2 x\right) + 3} + \sqrt{\left(x^{2} - 3 x\right) + 2} = - \frac{2 x}{x^{2} - 2 x + 3} + \sqrt{x^{2} + 3 x + 2}$$
- No
$$\frac{2 x}{\left(x^{2} + 2 x\right) + 3} + \sqrt{\left(x^{2} - 3 x\right) + 2} = \frac{2 x}{x^{2} - 2 x + 3} - \sqrt{x^{2} + 3 x + 2}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{2 x}{\left(x^{2} + 2 x\right) + 3} + \sqrt{\left(x^{2} - 3 x\right) + 2} = - \frac{2 x}{x^{2} - 2 x + 3} + \sqrt{x^{2} + 3 x + 2}$$
- No
$$\frac{2 x}{\left(x^{2} + 2 x\right) + 3} + \sqrt{\left(x^{2} - 3 x\right) + 2} = \frac{2 x}{x^{2} - 2 x + 3} - \sqrt{x^{2} + 3 x + 2}$$
- No
so, the function
not is
neither even, nor odd