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GRAPHICAL INTERPRETATION OF
SENTENCES LIKE  f(x)= 0  and  f(x)> 0

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If you know the graph of a function  f ,
then it is very easy to visualize the solution sets of sentences like  f(x)=0  and  f(x)>0 ;
this section shows you how!

Recall that the graph of a function  f  is a picture of all its  (input,output)  pairs;
that is, it is a picture of all points of the form  (x,f(x)) .

In particular, the y-value of the point  (x,f(x))  is the number  f(x) .
If  f(x)>0 , then the point  (x,f(x))  lies ABOVE the x-axis.
If  f(x)=0 , then the point  (x,f(x))  lies ON the x-axis.
If  f(x)<0 , then the point  (x,f(x))  lies BELOW the x-axis.
These concepts are illustrated below.

The notation  P(x,f(x))  is a convenient shorthand for:   the point  P  with coordinates  (x,f(x))

point  P(x,f(x))  has  f(x)>0  point  P(x,f(x))  has  f(x)=0  point  P(x,f(x))  has  f(x)<0 


The graph of a function  f  is shown at right.
The SOLUTION SET of the inequality  " f(x)> 0 "  is shown in purple.
It is the set of all values of  x  for which  f(x)  is positive.
That is, it is the set of  x-values that correspond to the part of the graph ABOVE the x-axis.
 
The graph of a function  f  is shown at right.
The SOLUTION SET of the equation  " f(x)= 0 "  is shown in purple.
It is the set of all values of  x  for which  f(x)  equals zero.
That is, it is the set of  x-intercepts of the graph.
 
The graph of a function  f  is shown at right.
The SOLUTION SET of the inequality  " f(x)< 0 "  is shown in purple.
It is the set of all values of  x  for which  f(x)  is negative.
That is, it is the set of  x-values that correspond to the part of the graph BELOW the x-axis.
 
The graph of a function  f  is shown at right.
The SOLUTION SET of the inequality  " f(x) 0 "  is shown in purple.
It is the set of all values of  x  for which  f(x)  is nonnegative.
That is, it is the set of  x-values that correspond to the part of the graph that is either ON or ABOVE the x-axis.
 
The graph of a function  f  is shown at right.
The SOLUTION SET of the inequality  " f(x) 0 "  is shown in purple.
It is the set of all values of  x  for which  f(x)  is nonpositive.
That is, it is the set of  x-values that correspond to the part of the graph that is either ON or BELOW the x-axis.

EXAMPLE

The graph of a function  g  with domain  [-6,10 )  is shown below:


Pay attention to the difference between the brackets "[ , ]" and parentheses "( , )" and braces "{ , }" in the following solutions sets!

The solution set of the inequality  " g(x)> 0 "  is:   (-3,-2 )(0,1 )(3,5 )[6,7 )(9,10 )

The solution set of the inequality  " g(x) 0 "  is:   (-3,-2] (0,1] [3,5] [6,7] [9,10)

The solution set of the equation  " g(x)= 0 "  is:   { -2, 1, 3, 5, 7, 9}

The solution set of the inequality  " g(x)< 0 "  is:    [-6,-3] (-2,0] (1,3) (5,6) (7,9)

The solution set of the inequality  " g(x) 0 "  is:    [-6,-3] [-2,0] [1,3] [5,6) [7,9]



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