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Plotting a function graph/ x-3

Graphing y = x-3

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = x - 3
f(x)=x−3f{\left(x \right)} = x - 3 
f = x - 3
The graph of the function
02468-8-6-4-2-1010-2020
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−3=0x - 3 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=3x_{1} = 3
Numerical solution
x1=3x_{1} = 3
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x - 3.
−3-3
The result:
f(0)=−3f{\left(0 \right)} = -3
The point:
(0, -3)
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
1=01 = 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
lim⁡x→−∞(x−3)=−∞\lim_{x \to -\infty}\left(x - 3\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(x−3)=∞\lim_{x \to \infty}\left(x - 3\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 x - 3, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(x−3x)=1\lim_{x \to -\infty}\left(\frac{x - 3}{x}\right) = 1
Let's take the limit
so,
inclined asymptote equation on the left:
y=xy = x
lim⁡x→∞(x−3x)=1\lim_{x \to \infty}\left(\frac{x - 3}{x}\right) = 1
Let's take the limit
so,
inclined asymptote equation on the right:
y=xy = 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−3=−x−3x - 3 = - x - 3
- No
x−3=x+3x - 3 = x + 3
- No
so, the function
not is
neither even, nor odd
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