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Plotting a function graph/ 1/(-7*x+5)

Graphing y = 1/(-7*x+5)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          1    
f(x) = --------
       -7*x + 5
f(x)=15−7xf{\left(x \right)} = \frac{1}{5 - 7 x} 
f = 1/(5 - 7*x)
The graph of the function
02468-8-6-4-2-10105-5
The domain of the function
The points at which the function is not precisely defined:
x1=0.714285714285714x_{1} = 0.714285714285714
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:
15−7x=0\frac{1}{5 - 7 x} = 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 1/(-7*x + 5).
15−0\frac{1}{5 - 0}
The result:
f(0)=15f{\left(0 \right)} = \frac{1}{5}
The point:
(0, 1/5)
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
7(5−7x)2=0\frac{7}{\left(5 - 7 x\right)^{2}} = 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
−98(7x−5)3=0- \frac{98}{\left(7 x - 5\right)^{3}} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
x1=0.714285714285714x_{1} = 0.714285714285714
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞15−7x=0\lim_{x \to -\infty} \frac{1}{5 - 7 x} = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞15−7x=0\lim_{x \to \infty} \frac{1}{5 - 7 x} = 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 1/(-7*x + 5), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(1x(5−7x))=0\lim_{x \to -\infty}\left(\frac{1}{x \left(5 - 7 x\right)}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(1x(5−7x))=0\lim_{x \to \infty}\left(\frac{1}{x \left(5 - 7 x\right)}\right) = 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:
15−7x=17x+5\frac{1}{5 - 7 x} = \frac{1}{7 x + 5}
- No
15−7x=−17x+5\frac{1}{5 - 7 x} = - \frac{1}{7 x + 5}
- No
so, the function
not is
neither even, nor odd
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