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The concept of composite functions, which are created by combining two functions. It covers the definition of composite functions, their domains, and provides examples. Additionally, it discusses one-to-one functions and how to determine if a function is one-to-one using the horizontal line test or the analytic definition.
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L20 Composite Functions; One-to-one Functions
A composite function f D g (read as โ f composed with g โ) is defined by ( f D g )( ) x = f ( g x ( )).
The domain of f D g is the set of all real x in the
Example: Show a diagram for the composite function
( f D g )( ) x = f ( g x ( ))
Similarly we define: ( g D f )( ) x = g ( f ( )) x
Example: Let f ( ) x = x โ 5 and g x ( ) = x^2 โ 2. Find:
(a) ( f D g )(4)=
(b) ( g D f )(9) =
(c) Find the composite functions and their domains
( g D f )( ) x =
Domain:
Example: Find functions f and g such that f D g = h if
h x ( ) = ( x^3^ + x^2 +1)^4
2
h x x
Example: An oil spill in the ocean assumes a circular
shape with an expanding radius r given by r = t + 1 , where t is the number of minutes after the measurements are started and r is measured in meters.
(a) Find a formula that gives the area A of the circular region as a function of time t.
(b) What is the area at the beginning? ( t = )
(c) What is the area 3 minutes later? ( t = )
The inverse of a function is a function itself if and only if for each y in the range there is only one x in the domain. In other words, no two ordered pairs have the same second coordinates, that is, no horizontal line intersects the graph in more than one point.
The functions for which the inverses are also functions are called one-to-one****.
Horizontal Line Test
If each horizontal line intersects the graph of a function f in at most one point, then f is one-to-one.
Analytic Definition :
A function f is one-to-one if and only if, for each a and b in the domain of f ,
f a ( ) = f b ( ) โ a = b or
Example: Determine by using the analytic definition which of the following functions is/are one-to-one: f ( ) x = 3 x โ 2
f ( ) x = x
Example: Use the Horizontal Line Test to determine whether the function is one-to-one.
Note: A function, which is increasing/decreasing on an interval I , is one-to-one on I.
Example: Determine whether the function
4 2 2
x y x
is one-to-one.
