Functions on a Cartesian Plane

Key Questions

  • Answer:

    There is a procedure to graph a function.

    Explanation:

    • Define the domain and codomain

    • Find the intersection between function and x-axes:
      solve #f(x)=0#

    • Calculate the first derivative and its intersection with x-axes:
      #f'(x)=0#. This points are called extrema, geometrically represent the points where the tangent of the function is horizontal. This mean that the function reach its minimum or maximum or stationary points.

    • Calculate the second derivative and its intersection with x-axes:
      #f''(x)=0#. This points (inflection point) are points on a curve at which the curve changes from being concave to convex or vice versa.
      if #f''(x)>0 # the function is convex (is smiling)
      if #f''(x)<0 # the function is concave (is sad)

    [Some typical example of domain](https://socratic.org/questions/how-do-you-find-the-domain-and-range-of-y-2x-2-4x-5#587652) and [graph function](https://socratic.org/questions/how-do-you-graph-y-3sqrt-x-3-and-compare-it-to-the-parent-graph#577391).

  • Answer:

    See explanation below

    Explanation:

    #(x,y)# is a pair of real numbers. The meaning is:

    #(x,y)# is an ordered pair of numbers belonging to #RRxxRR=RR^2#. The first pair memeber belongs to the first set #RR# and the second belongs to second #RR#. Althoug in this case is the same set #RR#. Could be in other cases #RRxxZZ# or #QQxxRR#

    #(x,y)# has the meaning of an aplication from #RR# to #RR# in which to every element x, the aplication asingns the y element.

    #(x,y)# has the meaning of plane's point coordinates. The first x is the horizontal coodinate (abscisa) and second is the vertical coordinate (ordenate). Both are coordinates.

    #(x,y)# has the meaning of a complex number: x is the real part and y is the imaginary part: #x+yi#

    #(x,y)# has the meaning of a plane's vector from origin of coordinates

    etc...

    You will see that meaning of #(x,y)# could be whatever of above depending of context, but if you think a little bit, all meanings are quite similar

    Hope this helps

Questions