Line Tests for Functions |
Definition of Function
- symbol: f : X → Y
- spoken: f is a function from X to Y.
- meaning: For every input x in the set X, there exists exactly one output y = f (x) in the set Y.
- meaning in symbols: ∀ x ∈ X ( ∃ !y ∈ Y ( f (x) = y ))
For real-valued functions of one real variable, the meaning is, more specifically,
- meaning: For every real number x, there exists exactly one real number output y = f (x).
- meaning in symbols: ∀ x ∈ R ( ∃ ! y ∈ R ( f (x) = y ))
- line test: Every vertical line touches the graph of f (x) exactly once.
For real-valued functions of one real variable with domain A ⊆ R and codomain B ⊆ R, the meaning is, more specifically,
- meaning: For every real number x in the set A, there exists exactly one real number output y = f (x) in the set B.
- meaning in symbols: ∀ x ∈ A ⊆ R ( ∃ ! y ∈ B ⊆ R ( f (x) = y ))
- line test: Every vertical line of the form x = a, for a ∈ A ⊆ R, touches the graph of f (x) exactly once at a point with coordinate y = b, for b ∈ B ⊆ R.
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Definition of Onto
- words: f : X → Y is onto
- meaning: For every desired output y in the set Y, there exists at least one input x in the set X such that y = f.
- meaning in symbols: ∀ y ∈ Y ( ∃ (at least one) x ∈ X ( f (x) = y ))
For real-valued functions of one real variable, the meaning is, more specifically,
- meaning: For every real number desired output y, there exists at least one real number input x such that y = f (x).
- meaning in symbols: ∀ y ∈ R ( ∃ (at least one) x ∈ R ( f (x) = y ))
- line test: Every horizontal line touches the graph of f (x) at least once.
For real-valued functions of one real variable with domain A ⊆ R and codomain B ⊆ R, the meaning is, more specifically,
- meaning: For every real number desired output y in the set B, there exists at least one real number input x in the set A such that y = f (x).
- meaning in symbols: ∀ y ∈ B ⊆ R ( ∃ (at least one) x ∈ A ⊆ R ( f (x) = y ))
- line test: Every horizontal line of the form y = b, for b ∈ B ⊆ R, touches the graph of f (x) at least once at a point with coordinate x = a, for a ∈ A ⊆ R.
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Definition of One-to-one
- words: f : X → Y is one-to-one
- meaning: For every desired output y in the set Y, there exists at most one input x in the set X such that y = f.
- meaning in symbols: ∀ y ∈ Y ( ∃ at most one x ∈ X ( f (x) = y ))
For real-valued functions of one real variable, the meaning is, more specifically,
- meaning: For every real number desired output y, there exists at most one real number input x such that y = f (x).
- meaning in symbols: ∀ y ∈ R ( ∃ at most one x ∈ R ( f (x) = y ))
- line test: Every horizontal line touches the graph of f (x) at most once.
For real-valued functions of one real variable with domain A ⊆ R and codomain B ⊆ R, the meaning is, more specifically,
- meaning: For every real number desired output y in the set B, there exists at most one real number input x in the set A such that y = f (x).
- meaning in symbols: ∀ y ∈ B ⊆ R ( ∃ at most one x ∈ A ⊆ R ( f (x) = y ))
- line test: Every horizontal line of the form y = b, for b ∈ B ⊆ R, touches the graph of f (x) at most once at a point with coordinate x = a, for a ∈ A ⊆ R.
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Definition of One-to-one and Onto
- words: f : X → Y is both one-to-one and onto
- meaning: For every desired output y in the set Y, there exists exactly one input x in the set X such that y = f.
- meaning in symbols: ∀ y ∈ Y ( ∃ ! x ∈ X ( f (x) = y ))
For real-valued functions of one real variable, the meaning is, more specifically,
- meaning: For every real number desired output y, there exists exactly one real number input x such that y = f (x).
- meaning in symbols: ∀ y ∈ R ( ∃ ! x ∈ R ( f (x) = y ))
- line test: Every horizontal line touches the graph of f (x) exactly once.
For real-valued functions of one real variable with domain A ⊆ R and codomain B ⊆ R, the meaning is, more specifically,
- meaning: For every real number desired output y in the set B, there exists exactly one real number input x in the set A such that y = f (x).
- meaning in symbols: ∀ y ∈ B ⊆ R ( ∃ ! x ∈ A ⊆ R ( f (x) = y ))
- line test: Every horizontal line of the form y = b, for b ∈ B ⊆ R, touches the graph of f (x) exactly once at a point with coordinate x = a, for a ∈ A ⊆ R.
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