The highest point on the graph of is
extreme values
For us, the codomain is often and so we often ignore this until it becomes
important. (There are many reasons for it to become important). Thus, we often
think of our real-valued functions as three sets: domain, range, and pairs.
We would like to know how the range values are affected by the domain values.
- We are interested in the function values - the range values. But they are
controlled by the domain values.
- We ask questions about the function values, but the answers are in the domain.
Where is the maximum value of the function?
Translation: Which domain value results in the maximum value of the function?
Where is the minimum value of the function?
Translation: Which domain value results in the minimum value of the function?
A graph provides a global view of all of the pairs, which reveals many of the patterns, features, and characteristics we seek.
However, we must separate the graph from the function. The graph helps us answer the questions, but the graph doesn’t hold the answers. The answers are in the domain and range. The answers are not the points in the graph. The answers are the numbers in the domain and range.
The graph displays patterns in its points. These patterns help us analyze the function which has values in the range connected to numbers in the domain.
Maximum
Let be a function defined on the domain .
Then is the maximum value of on if
- There exists , such that . i.e., has to be a function value
- for all . i.e., all function values are less than or equal to .
If there is no such , then has no maximum value.
Minimum
Let be a function defined on the domain .
Then is the minimum value of on if
- There exists , such that . i.e., has to be a function value
- for all . i.e., all function values are greater than or equal to .
If there is no such , then has no minimum value.
ooooo=-=-=-=-=-=-=-=-=-=-=-=-=ooOoo=-=-=-=-=-=-=-=-=-=-=-=-=ooooo
more examples can be found by following this link
More Examples of Function Graphs