Download Math 209 page 19 #69

Math 209 page 19 #43
x2
F ( x)  2
x  3x  4
Solution:
x2
F ( x)  2
x  (4  (1)) x  4 * (1)
x2
F ( x)  2
x  4 x  (1) x  4 * (1)
x2
F ( x)  2
( x  4 x)   (1) x  4*(1) 
x2
F ( x) 
x( x  4)  (1)  x  4 
x2
F ( x) 
( x  4)  x  (1) 
x2
F ( x) 
( x  4)( x  1)
Domain is the set of all real numbers except ( x  4)( x  1)  0 .
Domain is the set of all real numbers except ( x  4)  0 or ( x  1)  0 .
Domain is the set of all real numbers except x  4  0 or x 1  0 .
Domain is the set of all real numbers except x  4 or
x  1.
Domain is the set x : x  4 or  4  x  1 or 1  x  .
We express as an interval notation.
Domain= (, 4)  (4,1)  (1,) .
We find the range of the function.
x2
There are three factors in F ( x) 
. Those factors are x  4 , x  2 , x  1.
( x  4)( x  1)
x  4  0  x  4
x  2  0  x  2
x 1  0  x  1
When x near 4 , function F ( x) 
y
x2
can be approximated as
( x  4)( x  1)
42
6
2


. The graph is
( x  4)(4  1) ( x  4) *3 x  4
x2
can be approximated as
( x  4)( x  1)
x2
x2
x2
y


. The graph is
(2  4)(2  1) (2) * 3
6
When x near 2 , function F ( x) 
When x near 1 , function F ( x) 
y
x2
can be approximated as
( x  4)( x  1)
1 2
3

. The graph is
(1  4)( x  1) 5( x  1)
x2
is continuous except 4,1 .
( x  4)( x  1)
x2
Thus we can sketch the graph of y 
as
( x  4)( x  1)
The graph of F ( x) 
Thus the variable y can be any real number so the range of F ( x) 
set of real numbers.
The range = R .
x2
is the
( x  4)( x  1)