Transcript One-to-One Functions

Inverse Functions
Inverse Functions Domain and Ranges swap
places.
 Examples:
1. Given Elements
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2. Given ordered pairs
3. Given a graph
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When the inverse of a function f is itself a
function, then f is said to be a one-to-one
function. That is f is one-to-one if, for any
choice of elements in the domain of f, the
corresponding values in the range are
unequal.
In other words for every x there is a unique y
and for every y there is a unique x.
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If every horizontal line intersects the graph of
a function f in at most one point, then f is
one-to-one.
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A function that is increasing over its domain
is a one-to-one function. A function that is
decreasing over its domain is a one-to-one
function.
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The inverse function of f is denoted by the
symbol f-1
Be careful! This symbol does not mean the
reciprocal of f or 1/f(x).
Domain of f  Range of f 1 Range of f  Domain of f 1
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A function and its inverse are symmetric with
respect to the line y = x.
f 1 ( f ( x))  x and f ( f 1 ( x))  x
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Do the composition of the two functions.
If the answer is x, the functions are inverses
of each other.
If not, they are not inverses of each other.
Be sure the functions are one-to-one first.
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Swap the order of the ordered pairs.
In other words, make the x the y value and
the y the x value
Plot these points.
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First change f(x) to y
Swap the x’s and y’s
Solve the equation for y
Put the symbol for inverse in for y
To make sure your answer is correct, do the
composition and see if you get x.
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Examples
More Examples
Book Example
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Remember that domain of the original
function is the range of the inverse function
and vice versa.
Find the domain of the inverse function in
order to find the range of the original
function.
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We do this so that the inverse can now be a
function.
The quadratic function can have its domain
restricted to either x > 0 or x < 0 and its
inverse is now a function. (Look at the
horizontal line test)
The demand for corn obeys the equation
P(x) = 300 – 50x, where p is the price per
bushel (in dollars) and x is the number of
bushels produced, in millions. Express the
production amount x as a function of the
price p.
Why would this be important for a producer to
know?
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