(b) What is the area at the beginning? (t = )
(c) What is the area 3 minutes later? (t = )
Inverse Relations and Inverse Functions
Recall that a relation is a set of all ordered pairs (x, y),
where x is an element from the domain of the relation and
y is the corresponding element from the range.
Thus, the inverse relation we defined as the set of all
ordered pairs ( y, x).
Example: Find the inverse of the fol lowing relations .
Which of the relations are functions? De termine whether
the inverse relations are functions.
Note: Not for every function the inverse relation is a
The inverse of a function is a function itself if and only if
for each y in the range there is only one x in the domain.
In other words, no two ordered pairs have the same second coordinates , that is, no horizontal line intersects
the graph at more than one point.
The functions for which the inverses are also functions
are called one-to-one.
Horizontal Line Test
If each horizontal line intersects the graph of a
function f in at most one point, then f is one-to-one.
Example: Use the Horizontal Line Test to determine
whether the function is one-to-one.
Note: A function which is increasing/decreasing on an
interval I is one-to-one on I.
Note: A quadratic function y = a(x − h)^2 + k (a ≠ 0)
is not one-to-one, but, when considered on the restricted
domain, for example, on interval [h,+∞), it is one-to-one.
Remember, that the inverse of a function f is also a
function if and only if f is one-to-one.
Let f be a one-to-one function. Then g is the
inverse function of f if
for all x in the domain of g;
for all x in the domain of f.
If g is the inverse function of f, then we write g as
f-1(x) and read: “f-inverse”.
Example: Determine whether the following functions are
inverses of each other:
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