Mister Exam

Graphing y = 2*x+1

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 2*x + 1
f(x)=2x+1f{\left(x \right)} = 2 x + 1
f = 2*x + 1
The graph of the function
02468-8-6-4-2-1010-5050
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:
2x+1=02 x + 1 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=12x_{1} = - \frac{1}{2}
Numerical solution
x1=0.5x_{1} = -0.5
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2*x + 1.
02+10 \cdot 2 + 1
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
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
2=02 = 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
0=00 = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
limx(2x+1)=\lim_{x \to -\infty}\left(2 x + 1\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
limx(2x+1)=\lim_{x \to \infty}\left(2 x + 1\right) = \infty
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2*x + 1, divided by x at x->+oo and x ->-oo
limx(2x+1x)=2\lim_{x \to -\infty}\left(\frac{2 x + 1}{x}\right) = 2
Let's take the limit
so,
inclined asymptote equation on the left:
y=2xy = 2 x
limx(2x+1x)=2\lim_{x \to \infty}\left(\frac{2 x + 1}{x}\right) = 2
Let's take the limit
so,
inclined asymptote equation on the right:
y=2xy = 2 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:
2x+1=12x2 x + 1 = 1 - 2 x
- No
2x+1=2x12 x + 1 = 2 x - 1
- No
so, the function
not is
neither even, nor odd