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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*).

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.

- Add 4.
- Divide by 2.

We can verify that this is the inverse of

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

- Divide by 3.
- Add 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.

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