Mister Exam
Graphing y = 3^tgx×arcsin(7x^4)
The solution
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tan(x) / 4\ f(x) = 3 *asin\7*x /
$$f{\left(x \right)} = 3^{\tan{\left(x \right)}} \operatorname{asin}{\left(7 x^{4} \right)}$$
f = 3^tan(x)*asin(7*x^4)
The graph of the function
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:
$$3^{\tan{\left(x \right)}} \operatorname{asin}{\left(7 x^{4} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = -80.074868892622$$
$$x_{2} = -51.8040154624648$$
$$x_{3} = 0$$
$$x_{4} = -29.8117702392131$$
$$x_{5} = -95.7890397178945$$
$$x_{6} = 36.1642345681226$$
$$x_{7} = -7.81966445428965$$
$$x_{8} = -73.7964262390778$$
$$x_{9} = 14.1723366037438$$
so we need to solve the equation:
$$3^{\tan{\left(x \right)}} \operatorname{asin}{\left(7 x^{4} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = -80.074868892622$$
$$x_{2} = -51.8040154624648$$
$$x_{3} = 0$$
$$x_{4} = -29.8117702392131$$
$$x_{5} = -95.7890397178945$$
$$x_{6} = 36.1642345681226$$
$$x_{7} = -7.81966445428965$$
$$x_{8} = -73.7964262390778$$
$$x_{9} = 14.1723366037438$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(3^{\tan{\left(x \right)}} \operatorname{asin}{\left(7 x^{4} \right)}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(3^{\tan{\left(x \right)}} \operatorname{asin}{\left(7 x^{4} \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(3^{\tan{\left(x \right)}} \operatorname{asin}{\left(7 x^{4} \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(3^{\tan{\left(x \right)}} \operatorname{asin}{\left(7 x^{4} \right)}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 3^tan(x)*asin(7*x^4), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{3^{\tan{\left(x \right)}} \operatorname{asin}{\left(7 x^{4} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{3^{\tan{\left(x \right)}} \operatorname{asin}{\left(7 x^{4} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{3^{\tan{\left(x \right)}} \operatorname{asin}{\left(7 x^{4} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{3^{\tan{\left(x \right)}} \operatorname{asin}{\left(7 x^{4} \right)}}{x}\right)$$
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:
$$3^{\tan{\left(x \right)}} \operatorname{asin}{\left(7 x^{4} \right)} = 3^{- \tan{\left(x \right)}} \operatorname{asin}{\left(7 x^{4} \right)}$$
- No
$$3^{\tan{\left(x \right)}} \operatorname{asin}{\left(7 x^{4} \right)} = - 3^{- \tan{\left(x \right)}} \operatorname{asin}{\left(7 x^{4} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$3^{\tan{\left(x \right)}} \operatorname{asin}{\left(7 x^{4} \right)} = 3^{- \tan{\left(x \right)}} \operatorname{asin}{\left(7 x^{4} \right)}$$
- No
$$3^{\tan{\left(x \right)}} \operatorname{asin}{\left(7 x^{4} \right)} = - 3^{- \tan{\left(x \right)}} \operatorname{asin}{\left(7 x^{4} \right)}$$
- No
so, the function
not is
neither even, nor odd