Function Values

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expected values

Functions are packages. They contain three sets: domain, range, and a set of pairs. Each pair in the set of pairs partners a domain number with a range number. This range partner is called the value of the function at the domain number.

We have several tools to help us think about the pairs. Our favorites are formulas and graphs.

$\blacktriangleright$ A formula gives us step-by-step instructions on how to calculate the function value for a given domain number. Formulas are exact tools. They allow us to calculate function values exactly. However, they are local tools. We can use formulas to get function values, but one at a time.

$\blacktriangleright$ A graph is a visual tool that shows us a picture of all of the pairs. It is a global tool. Because of this, a graph is inherently inaccurate. We can approximate, but we cannot get exact values.

We use formulas and graphs for different purposes. They provide different views of a function. On the other hand, they both essentially providing the same information, function values. And, that is what we want from them.

Our main activity in Precalculus and Calculus is examining function values and their “behavior”. We want to know the function values. We want to compare some function values to other function values and compare those values to their corresponding domain numbers. We have already seen that this examination leads to expectations. Sometimes these expectations materialize and we are satisfied.

Sometimes these expectations are surprisingly wrong. The function value is not what we were expecting. All scientists like such surprises. Such events are “interesting”.

What function value was I expecting?
Why was I expecting a particular value?
What went wrong?
How is the function behaving?

Of course, we would like to categorize these unfulfilled expectations and find language to describe them. We want to communicate exactly about expectations and their comparisons to actuality.

We use words like discontinuity and singularity.

While function notation communicates the values of functions, we’ll need something else to describe expected values. Limits will be our language expected values and behavior.

Learning Outcomes

In this section, students will

contemplate closeness.
describe function behavior.

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more examples can be found by following this link
More Examples of Broken Values