Inverses of Functions
Warm Up #1 – Find the
following.
a. h  m  x 
f x   7 x  1
g x   x  3
x
h x  
x 1
2
m x  
x 1
2
b. h  m  2 
c.  fg  x 
d .  fg  b 
g
e.   x 
f 
f .  f  g  x 
g .  g  f  x 
Answer to a
x
2
h  m x  

x 1 x 1
x x  1  2 x  1

x  1x  1
x  x  2x  2

x  1x  1
2
x2  x  2

x  1x  1
Answer to b
x
2
h  m x  

x 1 x 1
x x  1  2 x  1

x  1x  1
x 2  x  2x  2

x  1x  1
x 2  3x  2

x  1x  1
2

 2   3 2   2
h  m  2 
 2  1 2  1
462

 1 3
8

3
Answer to c
 fg x   7 x  1x
2

3
 7 x  x  21x  3
3
2
Answer to d
 fg  b   7 b    b 
3
2
 21 b   3
 7b  b  21b  3
3
2
Answer to e
g
x 3
1
  x  
, x
7x 1
7
f
2
Answer to f
 f  g x   7x
2

 3 1
 7 x  21  1
2
 7 x  20
2
Answer to g
g  f x   7 x  1
2
3
 49 x 2  14 x  1  3
 49 x 2  14 x  2
g  f 4  494  144  2
 4916  56  2
2
 784  56  2
 838
Warm Up #2 - Determine whether the
statement is true or false. Justify your answer.
If f  x   x  1 and g  x   6 x,
then  f  g  x    g  f  x .
 f  g x  6x 1
g  f x   6x  1
 6x  6
FALSE!
Warm Up #3: Graph & give the
domain & range.
 x  5, x  3

f ( x)  2,
3  x 1
 x  4, x  1

Answer on Next Slide
Warm Up #3: Graph & give the
domain & range.
 x  5, x  3

f ( x)  2,
3  x 1
 x  4, x  1

x y
-3
-4
-5
2
1
0
x
1 -3
2 -2
3 -1
4 0
D :  ,3   3,  
R : ,
Inverse of a relation



The inverse of the ordered pairs (x, y) is the
set of all ordered pairs (y, x).
The Domain of the function is the range of the
inverse and the Range of the function is the
Domain of the inverse.
Symbol:
1
f ( x)
In other words, switch the
x’s and y’s!
Example: {(1,2), (2, 4), (3, 6), (4, 8)}
Inverse:
2,1, 4,2, 6,3, 8,4
Function notation? What is really
happening when you find the
inverse?
Find the inverse of f(x)=4x-2
x
*4
-2
(x+2)/4
/4
+2
1

x  2
x  
So
f
4
4x-2
x
To find an inverse…

Switch the x’s and y’s.

Solve for y.

Change to functional notation.
Find Inverse: f ( x)  8 x  1
f ( x)  8 x  1
y  8x 1
x  8 y 1
8y  x 1
x 1
y
8
x 1
1
f x  
8
Find Inverse: f ( x)  8 x  2
f ( x)  8 x  2
y  8x  2
x  8y  2
8y  x  2
x2
y
8
x2
1
f 
8
3x  1
f
(
x
)

Find Inverse:
2
3x  1
f x  
2
3x  1
y
2
3y 1
x
2
3y 1  2x
3y  2x 1
2x 1
y
3
2x 1
f 1 
3
Find Inverse: f ( x)  x  4
2
f x   x  4
2
y  x 4
2
x y 4
2
y  x4
2
y  x4
1
f ( x)   x  4
Draw the inverse. Compare to the
line y = x. What do you notice?
 5,3
 4,2
 3,1
 2,1
 1,1
0,1
1,5
yx
3,5
2,4
1,3
1,2
1,1
1,0
5,1
Graph the inverse of the following:
x
y
0 -5
-3 -4
1 2
1 4
The function and
its inverse are
symmetric with
respect to the
y-axis.
Things to note..


The domain of
f
1
x 
is the range of f(x).
The graph of an inverse function can be found
by reflecting a function in the line y=x.
Check this by plotting y = 3x + 1 and
x  1 on your graphic calculator.
3
Take a look
Reflecting..5
y=3x+1
4
3
y=x
2
1
y=(x-1)/3
0
-5
-4
-3
-2
-1
-1
-2
-3
-4
-5
0
1
2
3
4
5
Find the inverse of the function.
f ( x)  x
x y
x 
2
 y
2
y  x2
Is the inverse also a function? Let’s look at the
2


f
x

x
graphs.
f 1 x    x
If f  x   x 2 ,
x  y2
x
y2
y x
Inverse
NOTE: Inverse is NOT
a function!
Horizontal Line Test

If a horizontal line only passes through
one point at a time, then the inverse of
the function will also be a function.
Composition and Inverses

If f and g are functions and
( f  g )( x)  g  f x  x,
then f and g are inverses of one
another.
!!!!!!!!!!!!!!!!!!!!!!
Example: Show that the following are
inverses of each other.
1
2
f x   7 x  2 and g x   x 
7
7
2
1
 f  g x   7 x    2
7
7
 x22
x
g  f x   1 7 x  2  2
7
2 2
 x 
7 7
x
The composition of each both produce
a value of x; Therefore, they are inverses
of each other.
7
Are f & g inverses?
f ( x)  x 3  4
g ( x)  3 x  4
 f  g x   

3
x4 4
 x44
3
g  f x   3
x
 3 x3
x
YES!
x3  4  4
You Try….

Show that
1
3
f x   4 x  3 and g ( x)  x 
4
4

are inverses of each other.
 f  g x   g  f x   x
Therefore, they ARE
inverses of each other.
Are f & g inverses?
 x2
 f  g x   3
2
 3 
 x22
x
f ( x)  3x  2
x2
g ( x) 
3
3x  2  2
g  f x  
3
3x

3
x
YES!