Absolute Value

An absolute value function, also known as a modulus function, is an algebraic function where the variables are placed inside the absolute value bars. The common form of representing this function is f(x) = |x|, where x is a real number. 

The function can be written as f(x) = a |x - h| + k, where:

A is the vertical stretch of the graph,

‘h’ shows the horizontal shift, and

‘k’ is the vertical shift to the graph of f(x) = |x|.
The graph of an absolute value function is always V-shaped, and it opens upwards if a > 0 and opens downwards if a < 0. 
 

To understand the concept of the absolute value function, we need to learn its special properties. Let’s look at some common properties of the absolute value (modulus) function:

Idempotent Property

According to this property, if we apply the absolute value function twice, the result stays the same.
For example:
| |x| | = |x|, for any real number x.

Non-Negativity

For any real number, the absolute value is always zero or positive.
|x| ≥ 0, where x is a real number.

Multiplicative Property

For any real numbers a and b, the absolute value of their product is the same as the product of their absolute values.
|ab| = |a| × |b|, for all real numbers a and b.

Positive Definiteness

The absolute value of a number can be zero only when the number itself is zero.
|x| = 0 ⇔ x = 0

Symmetry

The absolute value of a number and its opposite are the same, irrespective of its sign 
|–x| = |x|

Triangle Inequality

The absolute value of a sum is always less than or equal to the sum of the absolute values of individual terms.
|a + b| ≤ |a| + |b|

Some related forms are:

||a| – |b|| ≤ |a – b|

|a + b| ≤ |a + b|

For any real numbers a₁, a₂, ..., aₙ:
  |a₁ + a₂ + ... + aₙ| ≤ |a₁| + |a₂| + ... + |aₙ|
 

The graph of the absolute value function is a V-shape. For example, the graph of y = |x| has a V shape with its vertex at the origin (0, 0).

To plot it on a graph, we first need to break the function into two parts, depending on whether x is positive or negative. This enables us to simplify the expression inside the absolute value.

For y = |x|, we can write:

y = x when x ≥ 0 (a straight line going up from the origin)

y = –x when x < 0 (a straight line going down to the left from the origin)

Now, we draw both lines on the graph:

For x ≥ 0, plot y = x, a line with a slope of 1.

For x < 0, plot y = –x, a line with a slope of –1.

When both lines are drawn, we get a V-shaped graph.

This graph is symmetric about the y-axis, which indicates that:

• Here, both sides of the graph are mirror images. 
• The point where the graph changes direction is called the vertex, and it lies at the origin for y = |x|.
   

The absolute value of a complex number gives the distance of the number from the origin in the complex plane.

If z = x + iy, where x is the real part and y is the imaginary part, then the absolute value of z, written as |z|, is:

|z| = √(x² + y²)

Example:
Find the absolute value of z = 5 + 12i.

Solution:
|z| = √(5² + 12²)
|z| = √(25 + 144)
|z| = √169
|z| = 13

So, the distance of z = 5 + 12i from the origin is 13 units.
 

In calculus, we look into the continuity, differentiability, derivative, and integration of absolute value functions.

Continuity of the Absolute Value Function

A function f(x) is said to be continuous at x = a if:
limx→a f(x) = f (a)
lim x→a⁻ f(x) = f(a) = f(a) [Left hand limit]
lim x→a⁺ f(x) = f(a+) = f(a) [Right hand limit]

Example: Check continuity of y = |x| at x = 3
Solution:
lim x→3⁻ |x| = 3
lim x→3⁺ |x| = 3
f(3) = |3| = 3

Since all values are equal, y = |x| is continuous at x = 3.

Differentiability of the Absolute Value Function

A function is differentiable at a point x = c if:

f'(c) = lim h→0 [f(c + h) - f(c)] / h

Example: Check differentiability of y = |x| at x = 0

f'(0) = lim h→0 |h| / h

If h > 0 → |h| / h = 1
If h < 0 → |h| / h = -1

Left-hand and right-hand limits are not equal, so x = 0, that is y = |x| is not differentiable at that point.
 

The derivative of |f(x)| is:

d/dx |f(x)| = f'(x) if f(x) > 0

d/dx |f(x)| = -f'(x) if f(x) < 0

Example: Derivative of y = |x|

If x > 0 → d/dx |x| = 1
If x < 0 → d/dx |x| = -1

Integration of the Absolute Value Function

To integrate |x|, split the interval at the point where x = 0.

Example: Evaluate ∫– 55  |x| dx

We know:
|x| = -x when x < 0
|x| = x when x ≥ 0

So,
I = ∫₋₅⁰ |x| dx + ∫₀⁵ |x| dx
= ∫₋₅⁰ (−x) dx + ∫₀⁵ x dx

Now calculate each part:
I = −∫₋₅⁰ x dx + ∫₀⁵ x dx

We know that ∫ x dx = (x²)/2, so:
I = −[(x²/2)]₋₅⁰ + [(x²/2)]₀⁵

= −[ (0²)/2 − (−5)²/2 ] + [ (5)²/2 − (0)²/2 ]
= −[ 0 − 25/2 ] + [ 25/2 − 0 ]
= 25/2 + 25/2
= 25
 

Now that we have understood the meaning of the absolute value function, let’s learn about the absolute value equation of the form:
f(x) = a |x - h| + k
This equation tells us how the graph changes based on the values of a, h, and k.

• The value of ‘a’ decides how much the graph stretches or compresses vertically.
• The value of h represents the horizontal shift.
• The value of k represents the vertical shift.
• The value of (h, k) represents the vertex of the graph 

The vertex of the graph can be calculated by solving (x - h) = 0 to get x = h. Then substitute it in the original function to find f(x), which is the value of k.
 

The absolute value of a number represents how far it is from zero on the number line, regardless of its sign.  Absolute values are frequently used in real life to determine the whole distance, difference, or change. Here, a few real-life applications of absolute value. 

• The absolute value is used to determine the temperature change of a place over time.For example, if the temperature changed from – 5oC to 3oC, the temperature change is: |3 – (-5)| = |3 + 5| = 8oC

• In sports, coaches utilize this concept to determine how much a team won or lost.For example:The score of Team A is 85 points and Team B is 75, so the coaches determine the score difference as: |85 – 75| = 10.

• Navigation systems like GPS make use of absolute value to calculate the shortest distances.For example: When your sister is 5 km south, and you are 5 km north, the distance between you both will be: |5 − (−5)| = |10| = 10 km.