Question 197800
<font size = 9 color = "red">Edwin's explanation:</b></font>
<pre><font size = 4 color = "indigo"><b>
The expression on the right of

{{{f(x) = 5 / (x + 7)}}} 

contains a denominator {{{x+7}}}, which must
never = 0, so

{{{x+7<>0}}}

or 

{{{x<>-7}}}

which means x must never equal -7, for there is a
vertical asymptote at {{{x=-7}}}  The graph is

{{{drawing(400,400,-15,5,-10,10,
graph(400,400,-15,5,-10,10,( sqrt(-x-7.1)/sqrt(-x-7.1) )*( 5/(x+7) ) ), 
graph(400,400,-15,5,-10,10,( sqrt(x+7)/sqrt(x+7) )*( 5/(x+7) ) )
)}}}

It approaches but never touches the black asymptote drawn
below at x=-7:

{{{drawing(400,400,-15,5,-10,10,
graph(400,400,-15,5,-10,10,( sqrt(-x-7.1)/sqrt(-x-7.1) )*( 5/(x+7) ) ), 
graph(400,400,-15,5,-10,10,( sqrt(x+7)/sqrt(x+7) )*( 5/(x+7) ) ),
line(-7,-11,-7,11)
)}}}

Therefore it cannot have -7 in its domain, which is the
two intervals indicated on the x-axis below:

{{{drawing(400,400,-15,5,-10,10,
graph(400,400,-15,5,-10,10,( sqrt(-x-7.1)/sqrt(-x-7.1) )*( 5/(x+7) ) ), 
graph(400,400,-15,5,-10,10,( sqrt(x+7)/sqrt(x+7) )*( 5/(x+7) ) ),

line(-16,0,-7.2,0), line(-6.8,0,6,0), locate(-7.2,.4,O)

)}}}

or on this number line, 
<font size = 1>
<=============================0====================================>
-15 -14 -13 -12 -11 -10  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2
</font>
which is the same thing as the x-axis on 
the graph above:

That is two intervals, and in interval notation is written:

       (-oo, -7) U (-7, oo)

Edwin</pre>