There is another method we can use to find the inverse of a function.
Taking the inverse "reverses" *x* and *f* (*x*). Thus, in the original function,
substitute "*x*" for "*f* (*x*)" and substitute "*f*^{-1}(*x*)" for "*x*". Then,
solve for *f*^{-1}(*x*) using inverse operations in the usual manner.

*Example 1*: *f* (*x*) = .

- Substitute:
*x*= - Solve for
*f*^{-1}(*x*):5 *x*= 2( *f*^{-1}(*x*)) - 15 *x*+ 1= 2( *f*^{-1}(*x*))= *f*^{-1}(*x*)*f*^{-1}(*x*)= - Check:
*f*^{-1}(*f*(*x*)) =*f*^{-1}() = = = =*x*

*Example 2*: *f* (*x*) = , *x*≠1, *f* (*x*) > 0.

- Substitute:
*x*= - Solve for
*f*^{-1}(*x*):*x*(*f*^{-1}(*x*) - 1)^{2}= 1 ( *f*^{-1}(*x*) - 1)^{2}= *f*^{-1}(*x*) - 1= *f*^{-1}(*x*)= + 1 - Check:
*f*^{-1}(*f*(*x*)) =*f*^{-1}( = +1 = + 1 = (*x*- 1) + 1 =*x*.

Domain of

We can also find the inverse of a function by graphing. The inverse of a
function is a reflection of that function over
the line *y* = *x*. In other words, all points (*x*, *y*) = (*a*, *b*) become (*x*, *y*) = (*b*, *a*). The *x* and *y* coordinates of each point switch:

Inverse of a Graph

To find the inverse of a function, reflect the function over the line *y* = *x*.
Or, find several points on the graph of *y* = *f* (*x*), switch their *x* and *y*
coordinates, and graph the resulting points. Connect these points with a line
or curve that mirrors the line or curve of the original function.

*Example*: Find the inverse *y* = 2*x* - 1 by graphing:

Inverse of *y* = 2*x* - 1

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