2. Concepts & Objectives
Inverse Functions
Review composition of functions
Identify 1-1 functions
Find the inverse of a 1-1 function
3. Composition of Functions
If f and g are functions, then the composite function, or
composition, of g and f is defined by
The domain of g f is the set of all numbers x in the
domain of f such that f x is in the domain of g.
g f x g f x
4. Composition of Functions (cont.)
Example: Let and .
a) Find
b) Find
2 1f x x
4
1
g x
x
2f g
3g f
5. Composition of Functions (cont.)
Example: Let and .
a) Find
b) Find
2 1f x x
4
1
g x
x
2f g
3g f
4
2 4
2 1
g 2 2 4f g f g f
2 4 1 7
3 2 3 1 7f
4 4 1
7
7 1 8 2
g
6. Composition of Functions (cont.)
Example: Let and .
c) Write as one function.
2 1f x x
4
1
g x
x
f g x
7. Composition of Functions (cont.)
Example: Let and .
c) Write as one function.
2 1f x x
4
1
g x
x
f g x
f g x f g x
4
2 1
1x
8
1
1x
8 1 8 1
1 1 1
x x
x x x
9
1
x
x
8. One-to-One Functions
In a one-to-one function, each x-value corresponds to
only one y-value, and each y-value corresponds to only
one x-value. In a 1-1 function, neither the x nor the y can
repeat.
We can also say that f a = f b implies a = b.
A function is a one-to-one function if, for
elements a and b in the domain of f,
a ≠ b implies f a ≠ f b.
10. One-to-One Functions (cont.)
Example: Decide whether is one-to-one.
We want to show that f a = f b implies that a = b:
Therefore, f is a one-to-one function.
3 7f x x
f a f b
3 7 3 7a b
3 3a b
a b
12. One-to-One Functions (cont.)
Example: Decide whether is one-to-one.
This time, we will try plugging in different values:
Although 3 ≠ ‒3, f 3 does equal f ‒3. This means that
the function is not one-to-one by the definition.
2
2f x x
2
3 3 2 11f
2
3 3 2 11f
13. One-to-One Functions (cont.)
Another way to identify whether a function is one-to-one
is to use the horizontal line test, which says that if any
horizontal line intersects the graph of a function in more
than one point, then the function is not one-to-one.
one-to-one not one-to-one
14. Inverse Functions
Some pairs of one-to-one functions undo one another.
For example, if
and
then (for example)
This is true for any value of x. Therefore, f and g are
called inverses of each other.
8 5f x x
5
8
x
g x
8 10 810 5 5f
85 5 80
8
8
8
105g
15. Inverse Functions (cont.)
More formally:
Let f be a one-to-one function. Then g is the inverse
function of f if
for every x in the domain of g,
and for every x in the domain of f.
f g x x
g f x x
16. Inverse Functions (cont.)
Example: Decide whether g is the inverse function of f .
3
1f x x 3
1g x x
17. Inverse Functions (cont.)
Example: Decide whether g is the inverse function of f .
yes
3
1f x x 3
1g x x
3
3
1 1f g x x
1 1x
x
3 3
1 1g f x x
3 3
x
x
18. Inverse Functions (cont.)
If g is the inverse of a function f , then g is written as f -1
(read “f inverse”).
In our previous example, for , 3
1f x x
1 3
1f x x
19. Finding Inverses
Since the domain of f is the range of f -1 and vice versa, if
a set is one-to-one, then to find the inverse, we simply
exchange the independent and dependent variables.
Example: If the relation is one-to-one, find the inverse of
the function.
2,1 , 1,0 , 0,1 , 1,2 , 2,2F not 1-1
3,1 , 0,2 , 2,3 , 4,0G 1-1
1
1,3 , 2,0 , 3,2 , 0,4G
20. Finding Inverses (cont.)
In the same way we did the example, we can find the
inverse of a function by interchanging the x and y
variables.
To find the equation of the inverse of y = f x:
Determine whether the function is one-to-one.
Replace f x with y if necessary.
Switch x and y.
Solve for y.
Replace y with f -1x.
21. Finding Inverses (cont.)
Example: Decide whether each equation defines a one-
to-one function. If so, find the equation of the inverse.
a) 2 5f x x
22. Finding Inverses (cont.)
Example: Decide whether each equation defines a one-
to-one function. If so, find the equation of the inverse.
a) one-to-one
replace f x with y
interchange x and y
solve for y
replace y with f -1x
2 5f x x
2 5y x
2 5x y
2 5y x
5
2
x
y
1 1 5
2 2
f x x
23. Graphing Inverses
Back in Geometry, when we studied reflections, it turned
out that the pattern for reflecting a figure across the line
y = x was to swap the x- and y-values.
It turns out, if we were to graph our inverse functions,
we would see that the inverse is the reflection of the
original function across the line y = x.
This can give you a way to check your work.
, ,x y y x
24. Graphing Inverses
2 5f x x
3
2f x x
1 1 5
2 2
f x x
1 3
2f x x