Line Tests

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.
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.
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.
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.

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.

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.

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.

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.