Analysis

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rigor

The goal of Precalculus is learning how to analyze functions. We are beginning with the Elementary Functions and then functions built from them.

What does it mean to analyze a function?

Analyzing a function means listing its characteristics, features, and aspects along with an explanation of how you decided on these characteristics, features, and aspects.

We want BOTH:
$\blacktriangleright $ A description of the characteristic
$\blacktriangleright $ Your reasoning on how you decided

Rigor

$\blacktriangleright $ Rigor lives inside your reasoning.

$\blacktriangleright $ Rigor lives inside your explanations.

Rigor means that you are using proper and precise mathematical language to explain how you know you are correct. This includes explaining how you know you have accounted for everything.

The List

The list of function characteristics, features, and aspects doesn’t change for our analysis.

Domain

For us, just starting out, the domain is a list of all real numbers that have been paired with a function value (another real number).

The domain might be stated as part of the function definition. The domain might be the natural domain implied by a formula. The domain might be the collection of first coordinates from points on a graph. The domain might be restricted by situational constraints. Your reasoning explains how you decided.

Domains are usually described with interval notation.

Zeros

Function zeros are domain numbers where the function value is $0$.

Discontinuities and Singularities

Discontinuities are domain numbers around which the function is behaving “wierd”. For us, just starting out, “weird” means that close domain numbers do not have close function values.

Singularities are non-domain numbers around which the function is behaving “wierd”.

We will need to invent some language to describe this “weird” behavior. Our language will be called limits.

Continuity

In Algebra, functions are continuous over domain intervals. For us, just starting out, we generally identify discontinuities and singularities, remove those, and are left with domain intervals where the function is continuous.

We will being with the idea that contunuity is where there are no discontinuities or singularities. Later, we will work on language to describe continuity.

End-Behavior

The end-behavior of a function is a description of how the function values behave as the domain values tend to $-\infty $ or $\infty $, the tails of the domain.

Our language for describing end-behavior will be called limits.

Behavior (Increasing and Decreasing)

The behavior of a function is a description of how the function values change with respect to changes in the domain. This includes where in the domain the function increases and decreases.

We will evetually extend this to more detailed descriptions around discontinuities and singularities.

We will use rates of change to measure function behavior.

Extrema

The extreme values of a function include the global maximum and global minimum values. These are also called the absolute maximum and absolute minimum values.

Extrema also includes local maximum and local minimum values. These are also called relative maximum and relative minimum values.

Along with the maximum and minmum function values, we want to know where in the domain these occur.

Range

The range is the collection of function values.

The range is usually described with interval notation.

Analysis (Version 1)

Let $N(z)$ be a function. The graph of $y = N(z)$ is displayed below.

Domain: The domain of $N$ is $(-\infty , 8]$.

Zeros: $-7$ and $0$ are the zeros of $N$,

Continuity: $-3$, $2$, and $8$ are discontinuities of $N$. There are no singularities.
$N$ is continuous on the intervals

\[ (-\infty , -3), (-3, 2), \, \text {and} \, (2, 8) \]

End-Behavior: $N$ becomes unbounded positively as the domain becomes unbounded negatively.

Behavior:

$N$ is decreasing on $(-\infty , -4.5)$.
$N$ is increasing on $(-4.5, -3)$.
$N$ is decreasing on $(-3, 2)$.
$N$ is decreasing on $(2, 8)$.

Global Maximum and Minimum: The end-behavior tells us that $N$ has no global maximum. The global minimum is $-8)$, which occurs at $8$.

Local Maximum and Minimum:

$N$ has a local minimum of $-8$, which occurs at $8$, because the global minimum is automatically a local minimum.
$N$ has a local minimum of $-6$, which occurs at $-4.5$.
$N$ has a local maximum of $5$, which occurs at $-3$.

Range: The range of $N$ is $\{ -8 \} \cup [-6, \infty )$.

Most likely, you can connect the analysis statements to the graph. That is not enough. The analysis above is not what we want. It is listing facts and leaving it to the reader to decide how those decisions were made. That isn’t what we want in the analysis of a function.

An analysis tells the reader how the author made the decisions.

Analysis (Version 2)

Let $N(z)$ be a function. The graph of $y = N(z)$ is displayed below.

Domain: The domain of $N$ is $(-\infty , 8]$. This is the collection of all of the first coordinates of the points on the graph.

Zeros: $-7$ and $0$ are the zeros of $N$. These are the $z$-coordinates of the $z$-intercepts.

Continuity: $-3$, $2$, and $8$ are discontinuities of $N$.

$3$ is a jump discontiuity, since $3$ is in the domain and there is space between the corresponding point and either side of the graph near this point.
$2$ is a jump discontiuity, since $2$ is in the domain and there is space between the corresponding point and the graph to the left of this point.
$8$ is a jump discontiuity, since $8$ is in the domain and there is space between the corresponding point and the graph to the left of this point.

These jumps are the only “weird” places in the graph, so there are no singularities.

$N$ is continuous on the intervals

\[ (-\infty , -3), (-3, 2), \, \text {and} \, (2, 8) \]

End-Behavior: $N$ becomes unbounded positively as the domain becomes unbounded negatively. The domain does not become unbounded to the right, so there is only left end-behavior.

Behavior:

$N$ is decreasing on $(-\infty , -4.5)$, because the graph goes down to the right.
$N$ is increasing on $(-4.5, -3)$, because the graph up down to the right.
$N$ is decreasing on $(-3, 2)$, because the graph goes down to the right.
$N$ is decreasing on $(2, 8)$, because the graph goes down to the right.

Global Maximum and Minimum: The end-behavior tells us that $N$ has no global maximum. The global minimum is $-8$, which occurs at $8$, because $(8,-8)$ is the lowest point on the graph.

Local Maximum and Minimum:

$N$ has a local minimum of $-8$, which occurs at $8$, because the global minimum is automatically a local minimum.
$N$ has a local minimum of $-6$, which occurs at $-4.5$, because $(-4.5, -6)$ is the lowest point on the interval $(-5, -4)$.
$N$ has a local maximum of $5$, which occurs at $-3$, because $(-3, 5)$ is the highest point on the interval $(-3.5, -2.5)$.

Range: The range of $N$ is $\{ -8 \} \cup [-6, \infty )$. This is the collection of all second coordinates of points on the graph.

As we move toward Calculus, we will discover that we talk about two types of increasing (and two types of decreasing).

In algebra, we talk about increasing over a domain interval.

As we move toward Calculus, we will extend this idea to increasing at a single domain number.

It will take some experience to recognize the context. So, we are usually forgiving with the endpoints.

In the analysis above we might say...

$N$ is decreasing on $(-\infty , -4.5]$, because the graph goes down to the right.
$N$ is increasing on $[-4.5, -3]$, because the graph up down to the right.
$N$ is decreasing on $[-3, 2]$, because the graph goes down to the right.
$N$ is decreasing on $[2, 8]$, because the graph goes down to the right.

...or, open intervals are just fine.

Note: $2$ is not the location of a local maximum or local minimum of $N$. That is because FOR EVERY open domain interval around $2$, there is ALWAYS a domain number to the left and right where the function value is greater than or less than $N(2)$. We can tell this is true, because the graph has points above and below the point $(2, -5)$ for EVERY POSSIBLE open domain interval around $2$.

Algebraic vs. Graphical Reasoning

We want precise analysis. We want to be exact.

The only way to do this is by using algebra. Graphs are inherently inaccurate and there is nothing we can do about this.

Our goal is rigorous algebraic reasong.

However, we are just starting out. We don’t have all of our algebraic tools yet. That is what this course is introducing. So, sometimes we have to provide graphical reasoning. It will be approximate rather than precise. We’ll just have to understand this. When we can be precise with algebra, then that is what we want.

We want precise analysis, which comes through logical reasoning and algebraic calculations, when we can get it.

We want to explain how we know we are exactly correct, but will settle for graphical feelings of correct, if that is all we can get.

Learning when the algebra will not produce the reasoning we want is part of learning mathematics.

Algebra and function reasoning first.

Then graphical reasoning.

But, always some reasoning. Always an explanation. Not just a declaration of facts.

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more examples can be found by following this link
More Examples of Visual Behavior

2025-08-02 18:20:30