Graphical Interpretation of Sentences like f(x)=0 and f(x)>0

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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]

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.

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

EXAMPLE

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