 # 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 one-to-one.

• example : Does y = x2 have an inverse?

• example : Does y = x2 , x ≥ 0 have an inverse?

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