Composite and Inverse Functions
3rd
4th Block
Composite Functions
• In a composition of functions, the range of
one function is part of the domain of the
other function
• Basically substituting one function into
another function
• Notation of composite functions is
• f(g(x)) or (f◦g)(x)
• Read as “f of g of x”
Composite Functions
f ( x)  x  4 x  3 and g ( x)  2 x  1
2
h( x )  4 x
find ...
f ( g ( x)) 
g ( f ( x)) 
( g  h)( x) 
( g  h  f )(2) 
Composite Functions
f ( x)   x  2 x and g ( x)  3 x  1
2
h( x )  x
find ...
f ( g ( x)) 
g ( f ( x)) 
( g  h)( x) 
(h  f )(1) 
Inverse Functions
• The domain of the inverse is the range of the
original function
• The range of the inverse is the domain of the
original function
• Switching x and y
To determine whether the inverse is a
function…
• Switch x and y values and determine
whether the domain of inverse is paired
with only one value in the range (domain
can not repeat)
Determine the inverse of the relation and decide whether it is a function
Relation : (2,3) (4,2) (5,2) (-3,3) (3,3)
Inverse :
Determining whether the inverse is a
function from the graph of the original
• You do the vertical line
test to determine
whether the graph
represents a function
• To determine whether
its inverse would be a
function, then you
would do a horizontal
line test
Finding the Inverse of a Relation
What is the inverse
of relation s?
Relation s
X
Y
0
-1
2
0
3
2
4
3
To find the inverse of a function
•
•
1.
2.
3.
4.
Notation: f -1(x)=
Steps:
Replace f(x) with y
Switch the x and y variable
Solve for y
Replace y with f -1(x)=
Find the following
f ( x)  2 x 2  3
g ( x)  5 x  2
h( x )  x  1
2x
m( x ) 
3
f 1 ( x) 
g 1 ( x) 
h 1 ( x) 
m 1 ( x) 
f 1 (2) 
Find the following
f ( x)  4 x 2  1
g ( x)   x  1
h( x )  x  2
x
m( x ) 
5
f 1 ( x) 
g 1 ( x) 
h 1 ( x) 
m 1 ( x) 
f 1 (3) 
Inverse functions
if
f ( g ( x))  x and g ( f ( x))  x
then the two functions are inverses
f ( x)  5 x  10
are they inverses ?
1
g ( x)  x  2
5
Are They Inverses?
• 𝑓 𝑥 = 4𝑥 + 6
• 𝑔 𝑥 =
𝑥−6
4
Are They Inverses?
• 𝑓 𝑥 = −6𝑥
• 𝑔 𝑥 =
1
𝑥
6