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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-
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. One-to-One Functions
Draw the graph of y = x2. 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 one-to-one function:
• A function f is one-to-one 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 one-to-one.
Ex: Which of the following functions has an inverse
(a) f (x) = |x|
(b) g (x) = x3
IV. Finding Inverse Functions Algebraically
Steps for finding the inverse of a function:
If the function is one-to-one
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