1. Chapter 7 Absolute Value and Reciprocal Funtions

2. 7.1 Absolute Value
The absolute value of a real number is its distance from zero on a real number line. For any real number x, the absolute value of x is written as |x|. The absolute value of a number will always be a positive number.
Example:
The absolute value of a positive number is positive.
| +9 | = 9
The absolute value of zero is zero.
|0| = 0
The absolute value of a negative number is the negative of that number, causing it to become positive
| -4 | = -( -4 )
= 4

3. 7.2 Absolute Value Functions
An absolute value function is a function that involves the absolute value of a variable. Any function of the form y = |f (x)| is called an absolute value function.
Since the function is defined by two different rules for each interval in the domain, you can define
y = |x| as the piecewise function. A piecewise function is a function made up of two or more separate functions with its own domain.
Ex. y = { x, if x ≥ 0 } , y = { -x, if x < 0 }
An invariant point is when the x-intercept of the original function is the x-intercept of the corresponding absolute value function.

4. Example: Consider the absolute value function. y = |2x - 3|
Example: Consider the absolute value function y = |2x - 3| . Determine the the y and x-intercept, state the domain and range, and express as a piecewise function.
Solve for y-intercept Solve for x-intercept
y = |2x - 3| |2x - 3| = 0
y = |2 (0) - 3| x - 3 = 0
y = | - 3 | x = 3
y = (0,3) x = 3 (3,0)
Domain: { x | x ∈ R }
Range: { y | y ≥ 0, y ∈ R }
Piecewise Function:
y = { 2x - 3, if x ≥ 3 } y = { -(2x - 3), is x < 3 }

5. 7.3 Absolute Value Equations
An absolute value equation is an equation that includes the absolute value of an expression involving a variable.
Ex. Solve |x - 3| = 7
Use the piecewise definition of the absolute value x - 3 = 0 x = 3
|x - 3| = { x - 3, x ≥ 3 } and { -(x - 3), if x < 3 }
Thus, when x ≥ 3 , |x - 3| = x - 3 When x < 3 , |x - 3| = -(x - 3)
|x - 3| = 7 |x - 3| = 7
x - 3 = (x - 3) = 7
x = x + 3 = 7
-x = 4
x = - 4

6. 7.4 Reciprocal Functions
A reciprocal function is a function y = 1 , where f(x) = x.
f(x)
Example: Graph the functions y = x and y = 1 . x
The function y = x is a function of one degree, so its graph is a line.
The function y = 1 is the rational function.
The graph has two distinct branches. These branches are on
either side of the vertical asymptote (defined by the non-
permissible value of the domain of the rational function).
There is also the horizontal asymptote, which is defined by that
the value 0 is not in the range of the function.

7. As recalled, the invariant points are points where both the original function and the reciprocal function intersect.
To find the invariant point, look at where your original function equals 1 and (-1).
Ex. 2x + 5 = 1 2x + 5 = - 1
2x = x = - 6
x = x = - 3
y = 2(-2) y = 2(-3) + 5
y = 1 y = - 1
( - 2, 1 ) ( - 3, - 1 )
the y- coordinates of the points on the graph of the reciprocal function are the reciprocals of the y-coordinates of the points on the graph of y = f(x)
as the value of x approaches a non-permissible value, the absolute value of the reciprocal function becomes large.
as the absolute value of x becomes large, the absolute value of the reciprocal function approaches zero.