1. 1.9
Inverse Functions
f-1(x)
Inverse functions have symmetry
about the line y = x
Inverse of ordered pairs.
Ex. f(x) = {(1,4), (2,-4), (7,3)}
The inverse is:
f-1(x) = {(4,1), (-4,2), (3,7)}
Note that the x and y’s are switched.

2. Definition of the Inverse of a Function
f and g are inverse functions if
f(g(x)) = x and
g(f(x)) = x
Ex. Show that the following functions are
inverses of each other.
f(x) = 2x3 -1
f(g(x)) =
= x + 1 – 1
= x

3. y = x

4. One-to-one Function: A 1-1 function states
that for every x there is exactly one y and
for every y there is exactly one x.
If a function is 1-1, then
f(a) = f(b) implies that a = b
Ex. f(x) = x3 + 1
Ex. f(x) = x2 - x
a3 + 1 = b3 + 1
a2 – a = b2 - b
a3 = b3
There’s no way
to get a = b.
a = b

5. A 1-1 function must pass both the vertical and
horizontal line tests.
Pg. 96 please
To find an inverse of a function f, use the
following procedure.
1. Interchange the x’s and y’s
2. Solve the equation for y
3. Replace f with f-1 in the new equation

6. Ex. Find the inverse of
2x = 5 – 3y
3y = 5 – 2x

7. The graph of f(x) and f-1(x)2 1
y = x