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Plotting a function graph/ x^(-3/4)

Graphing y = x^(-3/4)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
        1  
f(x) = ----
        3/4
       x
f(x)=1x34f{\left(x \right)} = \frac{1}{x^{\frac{3}{4}}} 
f = x^(-3/4)
The graph of the function
02468-8-6-4-2-1010010
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{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:
1x34=0\frac{1}{x^{\frac{3}{4}}} = 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^(-3/4).
10\frac{1}{0}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
−34x74=0- \frac{3}{4 x^{\frac{7}{4}}} = 0
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
2116x114=0\frac{21}{16 x^{\frac{11}{4}}} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
x1=0x_{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→−∞1x34=0\lim_{x \to -\infty} \frac{1}{x^{\frac{3}{4}}} = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞1x34=0\lim_{x \to \infty} \frac{1}{x^{\frac{3}{4}}} = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x^(-3/4), divided by x at x->+oo and x ->-oo
lim⁡x→−∞1x74=0\lim_{x \to -\infty} \frac{1}{x^{\frac{7}{4}}} = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞1x74=0\lim_{x \to \infty} \frac{1}{x^{\frac{7}{4}}} = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
1x34=1(−x)34\frac{1}{x^{\frac{3}{4}}} = \frac{1}{\left(- x\right)^{\frac{3}{4}}}
- No
1x34=−1(−x)34\frac{1}{x^{\frac{3}{4}}} = - \frac{1}{\left(- x\right)^{\frac{3}{4}}}
- No
so, the function
not is
neither even, nor odd
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