The absolute value function, \(|x|\), gives the distance on the number line between a number, \(x\), and \(0\). Since distance cannot be negative, it appears that the function just “makes numbers positive”.
To do this, the absolute value function just returns nonnegative numbers unharmed, and makes negative numbers turn positive. Algebraically, this is accomplished through a piecewise defined function.
Technically speaking, \(|-3| = -(-3)\) and then \(-(-3) = 3\). The absolute value function negates negative numbers.
Note: Just because you see a negaitve sign doesn’t mean you have a negative number.
- \(|4| = 4\)
- \(|0| = 0\)
- \(|-\pi | = -(-\pi ) = \pi \)
- \(|\cos (\pi )| = |-1| = -(-1) = 1\)
- \(|\sin (\tfrac {\pi }{4})| = \tfrac {1}{\sqrt {2}}\)
- \(|-\sqrt {5}| = \answer {\sqrt {5}}\)
- \(|4-4| = \answer {0}\)
- \(\left |\frac {-4}{-3}\right | = \answer {\frac {4}{3}}\)
- \(|\cos (\tfrac {\pi }{2})| = \answer {0}\)
- \(|\sin (\tfrac {3\pi }{2})| = \answer {1}\)
- \(|\tan (\tfrac {3\pi }{4})| = \answer {1}\)
There is no arithmetic operation called “make positive”. If a number is negative, then you make it positive by negating it.
Negating is arithmetic. “Make positive” is not arithmetic.
Graph of \(y = A(t) = |t|\).
![[Picture]](absoluteValue-d32ef6963c97bfa12f4b1eb2cb54d9b3.svg)
Continuity
The absolute value function is a continuous function.
- On \((-\infty , 0)\) we have \(| x | = -x\), which is a linear function, which is continuous.
- On \([0, \infty )\) we have \(| x | = x\), which is a linear function, which is continuous.
Both sides match up at \(0\), since
That graph suggests that \(|x|\) decreases on \((-\infty , 0]\) and increases on \([0, \infty )\).
This is confirmed by the formula.
On \((-\infty , 0]\), \(|x| = -x\), which is a decreasing linear functions.
On \([0, \infty )\), \(|x| = x\), which is an increasing linear functions.
Using the \(iRoC\) to say where \(A(x) = | x |\) increasing and decreasing.
First, get rid of the absolute value bars.
- \(x < 0\) when \(|x| = \answer {-x}\)
- \(x > 0\) when \(|x| = \answer {x}\)
Each piece is a restricted linear function. We can get their \(iRoC\).
\(iRoC_{|x|}\) is negative on \((-\infty , 0)\) and positive on \((0, \infty )\).
\(| x |\) is decreasing on \((-\infty , 0)\) and increasing on \((0, \infty )\).
Absolute value is a continuous function, which is decreasing on \((-\infty , 0)\) and increasing on \((0, \infty )\).
That makes \(| 0 | = 0\) the global minimum.
Since the absolute value function is a increasing linear function on \((0, \infty )\), we know that
Therefore, the range is \([0, \infty )\).
What are the cricital numbers for \(A(x) = | x |\) ?
First, \(0\) is in the domain.
The previous examples shows that the \(iRoC_{|x|}\) is defined and nonzero everywhere except \(0\).
The only possible critical number is \(0\). We will show that there is no tangent line at \((0,0)\).
Suppose there is a tangent line to the graph of \(y = | x |\) at \((0,0)\). Then, it has to match the slope
on the left, which is \(-1\) and it has to match the slope on the right, which is \(1\). A line
cannot have two slopes.
There is not tangent line at \((0,0)\), which means there is no slope.
\(0\) is the only critical number.
Absolute functions are those functions that can be represented by formulas of the form
where \(A\), \(B\), \(C\), and \(D\) are real numbers, with \(A \ne B\) and \(B \ne 0\) is a nonzero real number.
Note: In the template for absolute value functions, There is a leading coefficient for
the function and there is a leading coefficient for the linear function inside the
absolute value bars.
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more examples can be found by following this link
More Examples of Elementary Functions