Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

The Absolute Value Function, and its Properties
One of the most used functions in mathematics is the absolute value function
...

Absolute Value Function The absolute value of a real number x, |x|, is
x
if x ≥ 0
−x if x < 0

|x| =

The graph of the absolute value function is shown below
y

x

Example 1
|2| = 2,
|−2| = − (−2) = 2
The absolute value function is used to measure the distance between two numbers
...
Thus, the
distance from −2 to −4 is |−2 − (−4)| = |−2 + 4| = |2| = 2, and the distance from −2 to 5 is
|−2 − 5| = |−7| = 7
...

Lemma 1
...

This equality can be verified by considering cases
...
Then xy is ≤ 0 and we have
|xy| = − (xy) = (−x) y = |x| |y|
...


Lemma 2 For any real number x, and any nonnegative number a, we have
|x| ≤ a
if and only if
−a ≤ x ≤ a
...
We consider the case
of x ≥ 0
...
Then we have
−a ≤ 0 ≤ x = |x| ≤ a
...
Conversely suppose x ≥ 0 and −a ≤ x ≤ a
...

The argument for the case x < 0 is similar, and left to the reader
...

There are again two cases to consider: x positive or x negative
...
If
|x| ≥ a, then we have
x = |x| ≥ a
On the other hand if x ≥ a, then we have |x| = x ≥ a
...

In the case where x < 0, if we have x ≤ −a, then
|x| = −x ≥ − (−a) = a
Note when x < 0 the condition x ≥ a cannot be true, since we are assuming x ≥ 0
...

The proof of this is again handled by considering the four possible cases determined by the
signs of x and y
...

Note: in any of the above inequalities the less than or equal relation can be replaced with strict
inequality
...


Example 2 Find the interval of real numbers which contains x, if x satisfies the condition |2x − 5| <
3
...


Example 3 How close must the number x be to 4 if |3x − 12| < 5
...

5
5
5/3

Thus, x must be within 5/3 of 4 if 3x is to be within 5 units of 12

---