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
)2)(())((
))(())((
...
4)(
12)(34)( 2
fhgxhg
xfgxgf
find
xxh
xxgandxxxf
Composite Functions
)1)(())((
))(())((
...
)(
13)(2)( 2
fhxhg
xfgxgf
find
xxh
xxgandxxxf
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)
:Inverse
(3,3) (-3,3) (5,2) (4,2) (2,3) :Relation
function a isit whether decide andrelation theof inverse theDetermine
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
• Notation: f -1(x)=
• Steps:
1. Replace f(x) with y
2. Switch the x and y variable
3. Solve for y
4. Replace y with f -1(x)=
Find the following
)2(
)()(
)()(
3
2)(1)(
25)(32)(
1
11
11
2
f
xmxh
xgxf
xxmxxh
xxgxxf
Find the following
)3(
)()(
)()(
5)(2)(
1)(14)(
1
11
11
2
f
xmxh
xgxf
xxmxxh
xxgxxf
Inverse functions
?
25
1)(105)(
))(())((
inversestheyare
xxgxxf
inversesarefunctionstwothethen
xxfgandxxgfif
Are They Inverses?
• 𝑓 𝑥 = 4𝑥 + 6
• 𝑔 𝑥 =𝑥−6
4
Are They Inverses?
• 𝑓 𝑥 = −6𝑥
• 𝑔 𝑥 =1
6𝑥