Absolute Value

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The Core Absolute Value Function

The Core absolute value function is a piecewsie defined function.

\[ |k| = \begin{cases} -k & \text {if $k<0$,}\\ k & \text {if $k\ge 0$}. \end{cases} \]

Graph of $y = |k|$.

The symbol for the Core absolute value function are two vertical bars surrounding the domain value.

The domain is $(-\infty , \infty )$.

Its only zero is $0$.

It is continuous.

It is unbounded positively in both directions.

It decreases on $(-\infty , 0)$ and increases on $(0, \infty )$.

It has a global minimum of $0$ at $0$ and no global maximum. This is also the only local extreme number.

Its range is $[0, \infty )$.

All other absolute value functions are compositions of linear funcitons with the Core absolute value function.

General Absolute Value Functions

Absolute value functions are those functions that can be represented by formulas of the form

\[ A \, |B \, x + C| + D \, \text { with } \, A \ne 0, B \ne 0 \]

These can be viewed as compositions.

Let $|t|$ be the Core absolute value function.
Let $L_{in}(u) = B \, u + C$ be a linear function.
Let $L_{out}(v) = A \, v$ be a linear function.

Then, our general absolute value function can be expressed as a composition.

\[ A \, |B \, x + C| + D = L_{out}(v) \circ |t| \circ L_{in}(x) \]

Now we can use the Chain Rule to establish whether an absolute value function increases or decreases.

Absolute Value

Graph of $y = P(t) = -3|2t+4|+5$.

The formula:

\[ P(t) = -3|2t+4|+5 = \begin{cases} (-3)(-(2t+4))+5 & \text {if $t < \answer {-2}$,}\\ -3(2t+4)+5 & \text {if $t \ge \answer {-2}$}. \end{cases} \]

Domain: The domain of every absolute value function is $(-\infty , \infty )$.

Zeros:

$-3|2t+4|+5 = 0$

$|2t+4| = \frac {5}{3}$

Either $2t+4 = -\frac {5}{3}$ or $2t+4 = \frac {5}{3}$

Either $t = \frac {1}{2} \left ( -\frac {5}{3}-4 \right )$ or $t = \frac {1}{2} \left (\frac {5}{3} - 4 \right )$

Continuity: Absolute value functions are continuous.

End-Behavior: The leading coefficient is negative, therefore $P$ becomes unbounded negatively in either direction.

Behavior:

We can view $P$ as a composition:

Let $|x|$ be the Core absolute value function.
Let $L_{in}(u) = 2 \, u + 4$, an increasing linear function.
Let $L_{out}(v) = -3 \, v + 5$, a decreasinglinear function.

\[ P(t) = -3|2t+4|+5 = L_{out}(v) \circ |x| \circ L_{out}(t)\]

$|x|$ switches from decreasing to increasing at $0$, so we need to locate where $L_{in} = 0$.

$L_{in}(v) = 2 \, u + 4 = 0$ at $v = -2$

On $\left ( -\infty , \frac {5}{3} \right )$ the values of $L_{in} < 0$ and the Core absolute value function is decreasing on $(-\infty , 0)$.

On $\left ( -\infty , \frac {5}{3} \right )$, the Chain Rule gives

\[ dec \circ dec \circ inc = inc \]

On $\left ( \frac {5}{3}, \infty \right )$ the values of $L_{in} > 0$ and the Core absolute value function is increasing on $(0, \infty )$.

On $\left ( \frac {5}{3}, \infty \right )$, the Chain Rule gives

\[ dec \circ inc \circ inc = dec \]

Just CHecking: This agrees with the graph.

Global Maximum and Minimum:

Since the leading coefficient is negative, $P$ has no global minimum value.

$P$ switches from increasing to decreasing at $-2$, which means that $P(-2) = 5$ is global maximum value.

Local Maximums and Minimums: $P(-2) = 5$ is the only local maximum value. There are no local minimum values.

Range: $P$ is continuous. $5$ is its global maximum value. $P$ is unbounded negatively. That gives a range of $(-\infty , 5]$.

Absolute Value Analysis

Domain: The domain of every absolute function is $(-\infty , \infty )$.

Zeros: Absolulte value functions can have 2 zeros, 1 zero, or no zeros.

Continuity: Absolute value functions are continuous.

End-Behavior: In both directions absolute values become unbounded with the same sign. The leading coefficient gives the sign.

Behavior (Increasing and Decreasing): Absolute value functions switch their behavior. The critical number is the real number that makes the inside equal to $0$.

The Chain Rule can tell you which side is increasing and which side is decreasing.

Global Maximum and Minimum: Absolute value functions dhave either a global maximum or a global minimum.

Local Maximums and Minimums: The global extrema is also the only local extrema.

Range: The range of an absolute value function is either $(-\infty , D]$ or it is $[D, \infty )$.

Reading Coeffients

Behavior

Formulas for absolute value functions look like

\[ abs(x) = A | B \, x + C | + D \]

where $A \ne 0$, $B \ne 0$, $C$, and $D$ are real numbers.

The critical number for an absolute value function is the domain number that makes the inside of the absolute value bars equal to $0$.

\[ B \, x + C = 0 \]

The absolute value function increases on one side of the critical number and decreases on the other. The sign of the leading coefficient determines which way it goes.

If $A > 0$, then decreasing switching to increasing.
If $A < 0$, then increasing switching to decreasing.

End-Behavior

The end-behavior is the same on both sides for an absolute value function. The sign of the leading coefficient determines which way it goes.

If the leading coefficent is positive, then

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

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

If the leading coefficent is negative, then

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

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

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2026-05-17 15:05:34