Inverse functions - many-to-one and one-to-many

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Intermediate Mathematics - Inverse functions - many-to-one and
one-to-many

Inverse functions - MANY-TO-ONE AND ONE-TO-MANY

By definition, a function is a relation with only one function value for
each domain value. That is "one y-value for each x-value". In practice,
this means that a vertical line will cut the graph in only one place. For
example, the circle x+ y= 1, which has centre at the origin and a radius of
1 unit, is a relation and not a function. However, rearranging this
equation gives

Each of these drawn separately is a function.

is the upper semicircle and

is the lower semicircle.

A_ many-to-one function_ is a function which has more than one domain
value for each function value. That is "more than one x-value for each
y-value". In practice this means that a horizontal line will cut the graph
of the function in more than one place. For example either of the
semicircles above is a many-to-one function.

A _one-to-one function_ is a function which has only one domain value for
each function value. That is, "one x-value for each y-value". In practice
this means that a horizontal line will cut the graph of the function in
only one place. For example the semi-circle in the previous paragraph has
domain [-1,1] . The quarter circle in the first quadrant,

with domain [0,1] is a one-to-one function. It is usually possible to
restrict the domain of a many-to-one function so it becomes a one-to-one
function.

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