Graphical Interpretation of Sentences like f(x) = g(x) and f(x)

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

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This section should feel remarkably similar to the previous one, graphical interpretation of sentences like f(x) = 0 and f(x) > 0 .
The current section is more general—to return to the previous ideas, just let g(x) be the zero function!

If you know the graphs of two functions f and g ,
then it is very easy to visualize the solution sets of sentences like f(x)=g(x) and f(x)>g(x) ;
this section shows you how!

Recall that the graph of a function f is a picture of all points of the form (x,f(x)) ,
and the graph of a function g is a picture of all points of the form (x,g(x)) .

In particular, the y-value of the point (x,f(x)) is the number f(x)
and the y-value of the point (x,g(x)) is the number g(x) .

If f(x)>g(x) , then the point (x,f(x)) lies ABOVE the point (x,g(x)) .
If f(x)=g(x) , then the graphs of f and g INTERSECT at this point.
If f(x)<g(x) , then the point (x,f(x)) lies BELOW the point (x,g(x)) .
These concepts are illustrated below.

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

P1(x,f(x)) and P2(x,g(x)) with f(x)>g(x)	P1(x,f(x)) and P2(x,g(x)) with f(x)=g(x)	P1(x,f(x)) and P2(x,g(x)) with f(x)<g(x)

for this value of x , the graph of f lies ABOVE the graph of g	for this value of x , the graphs of f and g INTERSECT	for this value of x , the graph of f lies BELOW the graph of g

The graphs of functions f and g are shown at right. The SOLUTION SET of the inequality " f(x)> g(x) " is shown in green. It is the set of all values of x for which the graph of f lies ABOVE the graph of g .

The graphs of functions f and g are shown at right. The SOLUTION SET of the equation " f(x)= g(x) " is shown in green. It is the set of all values of x for which the graphs of f and g INTERSECT.

The graphs of functions f and g are shown at right. The SOLUTION SET of the inequality " f(x)< g(x) " is shown in green. It is the set of all values of x for which the graph of f lies BELOW the graph of g .

The graphs of functions f and g are shown at right. The SOLUTION SET of the inequality " f(x)≥ g(x) " is shown in green. It is the set of all values of x for which the graph of f lies ON or ABOVE the graph of g .

The graphs of functions f and g are shown at right. The SOLUTION SET of the inequality " f(x)≤ g(x) " is shown in green. It is the set of all values of x for which the graph of f lies ON or BELOW the graph of g .

EXAMPLE

The graphs of two functions, each with domain ℝ , are shown below:
f is a parabola, shown in the black dotted pattern;
g is a cubic polynomial, shown in purple.
These two curves intersect at the points (-1,2) , (0,0.5) and (1,1) .

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

The solution set of the inequality " f(x)> g(x) " is:   (-∞,-1) ∪(0,1)

The solution set of the equation " f(x)= g(x) " is:   {-1, 0,1}

The solution set of the inequality " f(x)< g(x) " is:   (-1,0) ∪(1,∞)

The solution set of the inequality " f(x)≥ g(x) " is:   (-∞,-1] ∪[0,1]

The solution set of the inequality " f(x)≤ g(x) " is:   [-1,0] ∪[1,∞)

You can use GeoGebra to explore the ideas from this section.
(Please be patient. It may take a few minutes for GeoGebra to load. It's worth the wait!)

GeoGebra Worksheet: Graphical Interpretation of Sentences like f(x) = g(x) and f(x) > g(x)

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