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Inverse Functions
• Let
What happens when we compose f and g ?
Notice that one function ‘undoes’ the other, producing a
final answer of x in both cases.
We say that f and g are inverses of each other.
• Definition : Two functions f and g are inverses of each other if
and
• example : Verify that and are inverses of each other.
• Notation : The inverse of function f is denoted by
• In the example above, we could write
and
• We can also restate the definition of inverse functions as
and
• example : Verify that if , then
• Result : The graphs of f and
are reflections of each other about the line
y = x.
• example : Graph the functions f and in the
example above.
• How do we find the inverse of a function?
• The algorithm:
1. Write the equation y = f ( x ) that defines the function.
2. Interchange the variables x and y .
3. Solve for
• example : Find the inverse of y = 3x + 2
1. y = 3x + 2
2. x = 3y + 2 ( interchange x and y )
3. ( solve for )
So,
• example : Find the inverse of f (x) = 4 − 2x
so,
• Does every function have an inverse function?
• To answer this question, think about the fact that the
graphs of f and
are reflections of each
other about the line y = x.
• Horizontal Line Test : A function has an inverse if no
horizontal straight line intersects its graph
more than once. The function is said to be onetoone.
• example : Does y = x^{2} have an inverse?
• example : Does y = x^{2} , x ≥ 0 have an inverse?
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