Inverse Functions
One-to-one
Suppose f : A ⇥ B is a function. We call f one-to-one if every distinct
pair of objects in A is assigned to a distinct pair of objects in B. In other
words, each object of the target has at most one object from the domain
assigned to it.
There is a way of phrasing the previous definition in a more mathematical
language: f is one-to-one if whenever we have two objects a, c ⇤ A with
a ⌅= c, we are guaranteed that f(a) ⌅= f(c).
Example. f : R ⇥ R where f(x) = x2 is not one-to-one because 3 ⌅= 3
and yet f(3) = f(3) since f(3) and f(3) both equal 9.
Horizontal line test
If a horizontal line intersects the graph of f(x) in more than one point,
then f(x) is not one-to-one.
The reason f(x) would not be one-to-one is that the graph would contain
two points that have the same second coordinate – for example, (2, 3) and
(4, 3). That would mean that f(2) and f(4) both equal 3, and one-to-one
functions can’t assign two dierent objects in the domain to the same object
of the target.
If every horizontal line in R2 intersects the graph of a function at most
once, then the function is one-to-one.
Examples. Below is the graph of f : R ⇥ R where f(x) = x2. There is a
horizontal line that intersects this graph in more than one point, so f is not
one-to-one.
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Inverse Functions
One-to-one
Suppose f : A —* B is a function. We call f one-to-one if every distinct
pair of objects in A is assigned to a distinct pair of objects in B. In other
words, each object of the target has at most one object from the domain
assigned to it.
There is a way of phrasing the previous definition in a more mathematical
language: f is one-to-one if whenever we have two objects a, c e A with
a ~ c, we are guaranteed that f(a) $ f(c).
Example. f : IR —* JR where f(x) = x2 is not one-to-one because 3 ~ —3
and yet f(3) = f(—3) since f(3) and f(—3) both equal 9.
Horizontal line test
If a horizontal line intersects the graph of f(.x) in more than one point,
then f(z) is not one-to-one.
The reason f(x) would not be one-to-one is that the graph would contain
two points that have the same second coordinate — for example, (2,3) and
(4,3). That would mean that f(2) and f(4) both equal 3, and one-to-one
functions can’t assign two different objects in the domain to the same object
of the target.
If every horizontal line in JR2 intersects the graph of a function at most
once, then the function is one-to-one.
Examples. Below is the graph of f : JR —, R where f(z) = z2. There is a
horizontal line that intersects this graph in more than one point, so f is not
Explanation:
