.
Suppose that the function f is defined for all real numbers as follows.
f(x)= -x^(2)+10 if -4<=x<4
-5-x if x>=4
Graph the function f. Then determine whether or not the function is continuous.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
In each of the two domains, the function is defined as a polynomial.
Therefore, inside each domain the function is continuous (since a polynomial is
a continuous function, as you should know from Calculus).
Hence, to be a continuous function over the entire domain, the necessary and sufficient condition
for the given function is to have the limit at x= 4 from the left to be equal to its value at x = 4.
The limit at x = 4 from the left side is -x^2 + 10 at x ---> 4, i.e. -4^2+10 = -16+10 = -6.
The value of the function at x = 4 is -5 - 4 = -9.
-9 =/= -6, so we conclude that the given function is not continuous at x = 4.
At this point, we complete our reasoning and answered the question.
//////////////////////////////////////
Notice that the problem's formulation has a deficiency.
Indeed, it says that "the function f is defined for all real numbers as follows . . . ",
but in reality it is defined only on the union of two intervals [-4,4) and [4,).
This union is the whole interval [,), but it is not the set of all real numbers,
as the problem proclaims. Thus, the problems wording is not perfect and is not accurate.
By such signs, every professional Math reader always can unmistakably recognize
"problems" that were written by an unprofessional "in a garage on the knee."
By the way, the first three words in the problem "Suppose that the" are excessive.
They are not necessary and can be omitted. Then the problem's appearance will be better.
It is a rule of Math writing: remove everything that does not matter.
In Math, following this rule is expressing of respect to a reader.