Graphing y = x^(sin(x))

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        sin(x)
f(x) = x

f{\left(x \right)} = x^{\sin{\left(x \right)}}

f = x^sin(x)

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:

x^{\sin{\left(x \right)}} = 0

Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to x^sin(x).

0^{\sin{\left(0 \right)}}

The result:

f{\left(0 \right)} = 1

The point:

(0, 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} x^{\sin{\left(x \right)}} = \left(-\infty\right)^{\left\langle -1, 1\right\rangle}

Let's take the limit
so,
equation of the horizontal asymptote on the left:

y = \left(-\infty\right)^{\left\langle -1, 1\right\rangle}

\lim_{x \to \infty} x^{\sin{\left(x \right)}} = \infty^{\left\langle -1, 1\right\rangle}

Let's take the limit
so,
equation of the horizontal asymptote on the right:

y = \infty^{\left\langle -1, 1\right\rangle}

Inclined asymptotes

Inclined asymptote can be found by calculating the limit of x^sin(x), divided by x at x->+oo and x ->-oo

\lim_{x \to -\infty}\left(\frac{x^{\sin{\left(x \right)}}}{x}\right) = \left(-\infty\right)^{\left\langle -2, 0\right\rangle}

Let's take the limit
so,
inclined asymptote equation on the left:

y = \left(-\infty\right)^{\left\langle -2, 0\right\rangle} x

\lim_{x \to \infty}\left(\frac{x^{\sin{\left(x \right)}}}{x}\right) = \infty^{\left\langle -2, 0\right\rangle}

Let's take the limit
so,
inclined asymptote equation on the right:

y = \infty^{\left\langle -2, 0\right\rangle} 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^{\sin{\left(x \right)}} = \left(- x\right)^{- \sin{\left(x \right)}}

- No

x^{\sin{\left(x \right)}} = - \left(- x\right)^{- \sin{\left(x \right)}}

- No
so, the function
not is
neither even, nor odd

The graph