Operations with Functions
Other Methods of Finding Inverses
Finding the Inverse of a Function by Isolating f -1(x)
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)
:
5x = 2(f -1(x)) - 1 5x + 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 f -1 : x > 0 . Range of f -1 : f -1(x)≠1 .
Finding the Inverse of a Function by Graphing
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:
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 = 2x - 1
by graphing:
