Linear Analysis

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8 characteristics

Linear Function Analysis

What do we want to know when we analyze any function?

We want to know the

Domain
Zeros
Continuity
$\circ $ discontinuities
$\circ $ singularities
End-Behavior
Behavior
$\circ $ intervals where increasing
$\circ $ intervals where decreasing
Global Maximum and Minimum
Local Maximums and Minimums
Range
...and we would like a nice graph

We want all of this information for linear functions and we want exact information, not approximations.

Remember, we are beginning with graphical analysis, because that is a familiar jumping off point for students. But, graphs are inherently inaccurate tools. That isn’t what we want. We are taking our familiarity with graphical descriptions and moving them over to algebraic descriptions, because algebra is our exact tool.

Linear functions are our initial bridge to exactness.

Linear functions are those functions which can be described with formulas like

\[ L(x) = A \, x + B \]

Constant Functions

The leading coefficient, $A$, can be $0$ in our template for linear functions.

\[ L(x) = B \]

In this case, the linear function is a constant function. Constant functions are special types of linear functions. Their graphs are lines, like the graphs of all linear functions. They are just horizontal lines.

Domain

All linear functions are defined for all real numbers. Their natural domain is $\mathbb {R} = (-\infty , \infty )$.

If you can identify a function as linear, then you automatically know its domain.

Restricting: We can restrict the domain of any function. Restricting the domain just means that the stated domain is not all of the numbers that could be used in the formula. For linear functions, we will alert the reader by titling such functions as restricted linear functions.

Zeros

Every time we describe a characteristic for a linear function, we must separate constant functions from other linear functions.

Constant functions do not have zeros, unless the constant function is the zero function. In that case, every number in the domain is a zero of the function.

Otherwise, linear functions have exactly one zero.

The zero of $L(x) = A \, x + B $ is $\frac {-B}{A}$

Restricting: If we restrict the domain and remove the zero, then it is possible our linear function will not have a zero.

Continuity

Linear functions are continuous functions. They have no discontinuities or singularities.

End-Behavior

Again, if the linear function is a constant function, then the end-behavior is just that constant.

Otherwise, linear functions have opposite end-behavior on either side of the domain, which is given by the sign of the leading coefficient.

If the leading coefficient is positive,
- the function is unbounded negatively in the negative tail of the domain.
- the function is unbounded positively in the positive tail of the domain.
If the leading coefficient is negative,
- the function is unbounded positively in the negative tail of the domain.
- the function is unbounded negatively in the positive tail of the domain.

This is probably easier to see, then write.

Algebraic Description

We will need some symbols and notation to make it easier to write about end-behavior.

Limits

Our notation for end-behavior is be called limit notation.

$\blacktriangleright $ the function is unbounded negatively in the negative tail of the domain.

\[ \lim \limits _{x \to -\infty } L(x) = -\infty \]

$\blacktriangleright $ the function is unbounded positively in the positive tail of the domain.

\[ \lim \limits _{x \to \infty } L(x) = \infty \]

$\blacktriangleright $ the function is unbounded positively in the negative tail of the domain.

\[ \lim \limits _{x \to -\infty } L(x) = \infty \]

$\blacktriangleright $ the function is unbounded negatively in the positive tail of the domain.

\[ \lim \limits _{x \to \infty } L(x) = -\infty \]

Limit notation uses “lim” as a shorthand abbreviation for limit. Under the “lim” we indicate which tail we are talking about. Under the “lim”, $variable \to -\infty $ indicates it is the left tail. Under the “lim”, $variable \to \infty $ indicates it is the right tail. The function name (or possibly formula) is then written to the right followed by an equal sign and then an expression describing the end-behavior.

Note: People write linear forms in many ways, like $m \, x + b$ or $A \, x + B$ and there will be others. Therefore, we do not want to refer to $m$ or $A$ in our analysis. We don’t want to refer to the symbol. We want to refer to the symbol’s role.

We will say “leading coefficent” for $A$.

We will say “the constant term” for $B$.

Behavior

Increasing and Decreasing

The graphs above vividly suggests that linear functions are always increasing or always decreasing (unless they are constant).

If the leading coefficient is positive, (then the graph moves up to the right) then the linear function is an increasing function.
If the leading coefficient is negative, (then the graph moves down to the right) then the linear function is a decreasing function.

Increasing and decreasing refer to the rate of change.

Increasing is a positive rate of change. (The domain and function values change in the same way.)
Decreasing is a negative rate of change. (The domain and function values change in the opposite way.)

Maximums and Minimums

Global and Local
Absolute and Relative

Linear functions do not have global maximums or minimums.
Linear functions do not have local maximums or minimums.

Unless it is a constant function, a linear function is unbounded on either side of its domain.

Restricting: If we restrict the domain, then we might have a maximum or minimum.

Again

This is all true if the linear function is not a constant function.

If the linear function is a constant function, then the function only has one function value at every domain number. That makes the one function value a global and a local maximum that occur at every domain number.

Weirdly enough, that also makes the one function value a global and local minimum that occurs at every domain number.

Range

If the linear function is a constant function, then the range is just a single number $\{ B \}$.

Otherwise, all linear function have the same range, $(-\infty , \infty )$

Restricting: Restricting the domain will also restrict the range.

Analysis

Analyze $L(x) = 3 \, x - 5$

explanation

$L(x) = 3 \, x - 5$ matches the template for a linear function $L(x) = A \, x + B$.
Since the leading coefficient is not $0$, $L$ is a nonconstant linear function.

Domain

As a linear function, the domain of $L$ is $(-\infty , \infty )$.

Zeros

As a nonconstant linear function, $L$ has a single zero.

$L(x) = 3 \, x - 5 = 0$ when $x = \frac {5}{3}$.

Continuity

As a linear function, $L$ is continuous. There are no discontinuities or singularities.

End-Behavior

Since the leading coefficient is $3$, which is positive, $L$ becomes unbounded negatively in the negative tail of the domain, while becoming unbounded positively in the positive tail of the domain.

\[ \lim \limits _{x \to \infty } L(x) = \infty \]

\[ \lim \limits _{x \to -\infty } L(x) = -\infty \]

Behavior

Since the leading coefficient is $3$, which is positive, $L$ is an increasing function.

Global Minimum and Maximum

As a nonconstant linear function, $L$ has no global maximum or minimum.

Local Minimum and Maximum

As a nonconstant linear function, $L$ has no local maximums or minimums.

Range

As a nonconstant linear function, the range of $L$ is $(-\infty , \infty )$.

Rigor: Reasons are given for each conclusion. That is rigor.

Analysis

Analyze $g(t) = \frac {7 - 3 \, t}{4}$

explanation

Although the formula is not matching the template for a linear function $L(x) = A \, x + B$, we can rewrite $g(t)$ in the form $g(t) = -\frac {3}{4} t + \frac {7}{4}$, which does match the template.

Note: It is the fact that $g(t)$ CAN be written in the official template form that makes it a linear funciton.

Since the leading coefficient is not $0$, $g$ is a nonconstant linear function.

Domain

As a linear function, the domain of $g$ is $(-\infty , \infty )$.

Zeros

As a nonconstant linear function, $g$ has a single zero.

$g(t) = \frac {7 - 3 \, t}{4}$ when $t = \frac {7}{3}$.

Continuity

As a linear function, $g$ is continuous. There are no discontinuities or singularities.

End-Behavior

Since the leading coefficient is $-\frac {3}{4}$, which is negative, $g$ becomes unbounded positively in the negative tail of the domain, while becoming unbounded negatively in the positive tail of the domain.

\[ \lim \limits _{x \to \infty } g(x) = -\infty \]

\[ \lim \limits _{x \to -\infty } g(x) = \infty \]

Behavior

Since the leading coefficient is $-\frac {3}{4}$, which is negative, $g$ is a decreasing function.

Global Minimum and Maximum

As a nonconstant linear function, $g$ has no global maximum or minimum.

Local Minimum and Maximum

As a nonconstant linear function, $g$ has no local maximums or minimums.

Range

As a nonconstant linear function, the range of $g$ is $(-\infty , \infty )$.

Reasons: Reasons for each conclusion. The quickest reason is to cite the category of the function, because we know lots of characteristics for functions in each category. This makes it very helpful to study categories of functions.

Composition

The composition of two linear functions is again a linear function.

Let $f(x) = A \, x + B$ and $g(t) = C \, t + D$ be two linear functions.

Their compositions look like

\[ (f \circ g)(w) = f(g(w) = f(C \, w + D) \]

\[ = A (C \, w + D) + B = A C \, w + A D + B \]

The rate of change of the composition is the product of the rates of change of the component linear functions.

If $A > 0$ and $C > 0$, then $A C > 0$
If $A > 0$ and $C < 0$, then $A C < 0$
If $A < 0$ and $C > 0$, then $A C < 0$
If $A < 0$ and $C < 0$, then $A C > 0$

This gives us a map that connects the rates of change of linear functions to the rate of change of their composition, which we call the Chain Rule.

Chain Rule

\[ inc \circ inc = inc \]

\[ inc \circ dec = dec \]

\[ dec \circ inc = dec \]

\[ dec \circ dec = inc \]

ooooo-=-=-=-ooOoo-=-=-=-ooooo
more examples can be found by following this link
More Examples of Linear Functions

2026-06-06 15:11:09