Mister Exam

Graphing y = x^(sin(x))

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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        sin(x)
f(x) = x      
f(x)=xsin(x)f{\left(x \right)} = x^{\sin{\left(x \right)}}
f = x^sin(x)
The graph of the function
02468-8-6-4-2-1010010
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:
xsin(x)=0x^{\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).
0sin(0)0^{\sin{\left(0 \right)}}
The result:
f(0)=1f{\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
limxxsin(x)=()1,1\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=()1,1y = \left(-\infty\right)^{\left\langle -1, 1\right\rangle}
limxxsin(x)=1,1\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=1,1y = \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
limx(xsin(x)x)=()2,0\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=()2,0xy = \left(-\infty\right)^{\left\langle -2, 0\right\rangle} x
limx(xsin(x)x)=2,0\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=2,0xy = \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:
xsin(x)=(x)sin(x)x^{\sin{\left(x \right)}} = \left(- x\right)^{- \sin{\left(x \right)}}
- No
xsin(x)=(x)sin(x)x^{\sin{\left(x \right)}} = - \left(- x\right)^{- \sin{\left(x \right)}}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = x^(sin(x))