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Inverse Functions

Two functions *f* and *g* are inverse functions if *f*o*g*(*x*) = *x* and *g*o*f* (*x*) = *x* for all values of *x* in the domain of *f* and *g*.

For instance, *f* (*x*) = 2*x* and *g*(*x*) = *x* are inverse functions
because *f*o*g*(*x*) = *f* (*g*(*x*)) = *f* (*x*) = 2(*x*) = *x* and *g*o*f* (*x*) = *g*(*f* (*x*)) = *g*(2*x*) = (2*x*) = *x*. Similarly, *f* (*x*) = *x* + 1
and *g*(*x*) = *x* - 1 are inverse funcions because *f*o*g*(*x*) = *f* (*g*(*x*)) = *f* (*x* - 1) = (*x* - 1) + 1 = *x* and *g*o*f* (*x*) = *g*(*f* (*x*)) = *g*(*x* + 1) = (*x* + 1) - 1 = *x*.
*h*(*x*) = 3*x* - 1 and *j*(*x*) = are inverse functions because *h*o*j*(*x*) = *h*(*j*(*x*)) = *h*() = 3() - 1 = *x* + 1 - 1 = *x* and *j*o*h*(*x*) = *j*(*h*(*x*)) = *j*(3*x* - 1) = = = *x*.

The inverse of a function *f* (*x*) is denoted *f*^{-1}(*x*).

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Finding the Inverse of a Function by Reversing Operations

The trick to finding the inverse of a function *f* (*x*) is to "undo" all the
operations on *x* in reverse order.

The function *f* (*x*) = 2*x* - 4 has two steps:

- Multiply by 2.
- Subtract 4.

Thus,

*f*^{-1}(*x*) must have two steps:

- Add 4.
- Divide by 2.

Consequently,

*f*^{-1}(*x*) = .

We can verify that this is the inverse of

*f* (*x*):

*f*^{-1}(*f* (*x*)) = *f*^{-1}(2*x* - 4) = = = *x*.

*f* (*f*^{-1}(*x*)) = *f* () = 2() - 4 = (*x* + 4) - 4 = *x*.

*Example 1*: Find the inverse of *f* (*x*) = 3(*x* - 5).

Original function:

- Subtract 5.
- Multiply by 3.

New function:

- Divide by 3.
- Add 5.

Thus,

*f*^{-1}(*x*) = + 5.

Check:

*f*^{-1}(*f* (*x*)) = *f*^{-1}(3(*x* - 5)) = + 5 = (*x* - 5) + 5 = *x*.

*f* (*f*^{-1}(*x*)) = *f* ( +5) = 3(( +5) - 5) = 3() = *x*.