Domain of a Function

Domain Restrictions and Functions Defined Piecewise
An important concept in the study of functions, especially piece-wise defined functions, is that of domain restrictions. Domain restrictions allow us to create functions defined over numbers that work for our purposes. Piecewise defined functions are the composition of multiple functions with domain restrictions that do not overlap. Some functions are restricted from values that make them undefined.

Graphing Piecewise Defined Functions
How to graph piecewise defined function by hand and on the graphing calculator?
A piecewise-defined function (also called a piecewise function) is a function whose rule changes depending on the value of the input.

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Ex 1: Graph a Piecewise Defined Function.

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Ex 2: Graph a Piecewise Defined Function.

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Ex 3: Graph a Piecewise Defined Function.

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Finding the Domain of a Function
Certain functions, such as rational and radical elementary functions, have instances of restricted domains. When finding the domain of a function, we must always remember that a rational function involves removing the values that could make the denominator of a fraction zero. Finding the domain of a function that is radical means not making the radical negative.

How to find the domain of a function when it has rational or radical expressions?
How to find the domain of 3 different rational expressions that have quadratic expressions in the denominator?
Examples:
Find the domain:
a) (x - 2)/(x² -2x - 35)
b) x/(9x² - 3x)
c) (x³ + 4x)/(x⁴ - 1)

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Finding Domain and Range of a Function using a Graph
To find the domain form a graph, list all the x-values that correspond to points on the graph.
To find the range, list all the y values.
Examples:
Using interval notation, state the domain and range of each given graph.

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How to find the domain of a function, without graphing?
Example:

1. f(x) = 1/(x - 2)
2. f(x) = 1/(x² - x - 6)
3. f(x) = √(x - 1)/(x² + 4)
4. f(x) = 1/√(x² - 4)
5. f(x) = ln(x - 8)

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Finding the Domain of a Function Algebraically
Find the domain:
a) 1/(x² - 7x - 30)
b) g(x) = √(2x + 3)

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