connecting reals
Real-Valued Functions
For the most part, our attention in this course is focused on real-valued functions.
The values of a real-valued function are real numbers.Let the function be defined as follows.
- Domain of is .
- Codomain of is .
- pairs a domain number with its square.
First, this function is well-defined, since the square of any number in will be in and each domain number has exactly one square.
Shorthand Notation: .
is not an onto function. For instance, , yet is not in the range, since squares of real
numbers cannot be negative.
is not a one-to-one function, since .
Let the function be defined as follows.
- Domain of is the positive integers: .
- Codomain of is the positive integers: .
- pairs a domain number with a range number according to the following
rule:
- If the domain number is even, then pairs it with half the domain number:
- If the domain number is odd, then pairs it with one more than three times the domain number.
First, this function is well-defined, since the calculations can only produce one result.
Shorthand Notation: .
is not a one-to-one function since .
Let the function be defined as follows.
- Domain of is all real numbers: .
- Codomain of is all real numbers: .
- pairs a domain number with itself.
First, this function is well-defined.
Shorthand Notation: .
The Identity function is an onto function. If , the range, then .
The Identity function is a one-to-one function, since if , then . If two values are equal, then the domain numbers are equal. They were not different domain numbers.
Let the function be defined as follows.
- Domain of is all natural numbers: .
- Codomain of is .
- pairs a natural number with the remainder when divided by .
First, this function is well-defined. Dividing by can only have one remainder.
Shorthand Notation: .
Let the function be defined as follows.
- Domain of is .
- Codomain of is .
- pairs a number with twice the number.
First, this function is well-defined.
Shorthand Notation: .
Communication Summary
The range is also called the image of the function.
Sometimes the range partner of a domain number is called the image of the domain number.
is called the “value of at ” or “the image of under ”.
is pronounced “ of ”.
And, we have the reverse direction.
The preimage of a subset of the codomain consists of the domain members whose function values are inside the given subset.
The preimage of is the set
Notation: The exponent does not mean reciprocal. Instead, it is conveying an “opposite” direction.
The preimage of the range is the domain.
Sometimes when the preimage is a single domain member, then we drop the idea of a
set and just quote that one domain member.
Note: The preimage of a codomain number, which is not in the range, is the empty
set, .
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more examples can be found by following this link
More Examples of Real-Valued Functions