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Inverse Functions
I. Inverse Functions
If we form the composition of two functions , we should get
the identity function h (x) = x. So to speak, inverse
functions “undo” each other. Also note that if a function is
given as a set of ordered pairs , its inverse has all the x coordinates
interchanged with their corresponding y
coordinates .
Ex: f (x) = x + 2 and g (x) = x – 2 are inverses because
f (g (x)) = f (x  2) = (x  2) + 2 = x, and
g (f (x)) = g (x + 2) = (x + 2) – 2 = x.
The functions f and g are inverses of each other iff:
1. f (g (x)) = x for every x in the domain of g, and
2. g (f (x)) = x for every x in the domain of f.
In this case we write g (x) as f^{ 1}(x)
This is NOT a negative exponent it is just the notation
II. The Graph of an Inverse Function
If we interchanged the x’ s and y’ s for every point on the
graph of a function , the graph would be reflected about
the line y = x.
Ex: f (x) = x – 1 and f^{ 1}(x) = x + 1
III. OnetoOne Functions
Draw the graph of y = x^{2}. Reflect it about the line y = x.
Is this inverse a function.
When is the inverse of a function a function?
This brings us to the Horizontal Line Test :
• If every horizontal line meets the graph of a function
in at most one point, then the inverse of this function
will be a function.
• If no horizontal line intersects the graph of a function
in more than one point, no x value is matched with
more than one y value .
This brings us to the definition of a onetoone function:
• A function f is onetoone if each value of the
dependent variable corresponds to exactly one value
of the independent variable .
A function has an inverse iff the function is onetoone.
Ex: Which of the following functions has an inverse
function?
(a) f (x) = x
(b) g (x) = x^{3}
IV. Finding Inverse Functions Algebraically
Steps for finding the inverse of a function:
If the function is onetoone
1. Replace f (x) with y.
2. Interchange the roles of x and y.
3. Solve this new equation for y .
4. Replace y by f ^{1}(x).
Ex: Find the inverse of each of the following functions.
(c) h (x) = x ^{2} for x ≥ 0
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