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Understanding Functions: Definitions and Properties
Sep 1, 2024
Functions and Their Properties
Definition of a Function
A function is a relation where each element of the domain maps to exactly one element of the range.
Every x has only one y.
Example: If x1 maps to y1, x2 maps to y2, that forms a function.
A mapping diagram can help visualize if a relation is a function.
Example of a non-function: If x1 maps to both y1 and y3, it is not a function.
Determining Functions
Mapping Diagram
: Useful for visualizing whether each x maps to only one y.
Vertical Line Test
:
A visual test where a vertical line swept across the graph should not intersect it more than once at any location to confirm it's a function.
Domain and Range
Domain
: Set of all possible input values (x-values).
Range
: Set of all possible output values (y-values).
Domain and Range of Algebraic Functions
Square Roots
:
Cannot have a negative number under the square root in real numbers.
Example: For sqrt(x + 4), x + 4 >= 0 implies x >= -4.
Fractions
:
Cannot divide by zero.
Example: For 1/(x - 3), x cannot be 3.
Interval Notation
:
Domain: x ≥ -4 is represented as [-4, ∞).
Excluding 3: [0, 3) ∪ (3, ∞).
Range
:
Use graphing calculator for visualization.
Example: Exclude y = 0 range for functions crossing x-axis.
Contextual Domain and Range
Problem-specific restrictions:
Volume of a Sphere
:
Domain: Radius r cannot be zero or negative.
Range: Volume cannot be negative.
Domain and Range for such problems are often deduced using context and common sense.
Example: Domain and Range for volume of a sphere are both (0, ∞).
Key Points
A function is defined by its unique x-y mapping.
Domain and range are crucial for understanding the behavior of functions.
Use graphical methods and contextual insights to determine domain and range.
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