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Exponential and Logarithmic Functions

Composite
Functions

Section 4.1

Composite Functions

•Construct new function from two
given functions f and g

•Composite function:

•Denoted by  f o g
•Read as “fcomposed with g”
•Defined by



•Domain: The set of all numbers x in the
domain
of g such that g(x) is in the
domain of f.

•Note that we perform the inside
function g(x) first.


 

•Example. Suppose that f(x) = x3 - 2
and g(x) = 2x2+ 1. Find the values
of the following expressions .

(a) Problem: (f o g)(1)

Answer:

(b) Problem :(g o f)(1)

Answer:

(c) Problem: (f o f)(0)

Answer
:

•Example. Suppose that f(x) = 2x2+ 3 and
g(x) = 4x3+ 1.

(a) Problem: Find f o g.

Answer:

(b) Problem: Find the domain of f o g.

Answer:

(c) Problem: Find g o f.

Answer:

(d) Problem: Find the domain of f o g.

Answer:

Example. Suppose that and


(a) Problem: Find f o g.

Answer:

(b) Problem: Find the domain of f o g.

Answer:

(c) Problem: Find g o f.

Answer:

(d) Problem: Find the domain of f o g.

Answer:

•Example.

Problem: If f(x) = 4x+ 2 and
, show that for all x,

Decomposing Composite
Functions

• Example.

Problem: Find functions f and g such that

Answer:

Key Points

•Composite Functions

•Decomposing Composite Functions

One-to-One
Functions;
Inverse Functions


Section 4.2

One-to-One Functions

•One-to-one function: Any two
different inputs in the domain
correspond to two different outputs in
the range.

•If x1 and x2 are two different inputs of a
function f, then

•One-to-one
function
•Not a one-to-one
function
•Not a function

•Example.

Problem: Is this function one-to-one?

Answer:

•Example. Determine whether the
following functions are one-to-one.

(a) Problem: f(x) = x2+ 2

Answer:

(b) Problem: g(x) = x3- 5

Answer:

•Theorem.

A function that is increasing on an
interval I is a one-to-one function on
I.

A function that is decreasing on an
interval I is a one-to-one function on
I.

Horizontal- line Test

•If every horizontal line intersects the
graph of a function fin at most one
point, then f is one-to-one.

Example.

Problem: Use the graph to determine
whether the function is one-to-one .

Answer :

•Example.

Problem: Use the graph to determine
whether the function is one-to-one .

Answer :

Inverse Functions

•Requires f to be a one-to-one function

•The inverse function of f

•Written f -1
•Defined as the function which takes

•f(x) as input
•Returns the output x.

•In other words, f -1undoes the action of
f

•f -1(f(x)) = x for all x in the domain of f
•f(f -1(x)) = x for all x in the domain of f -1

•Example. Find the inverse of the
function shown

Problem:

Answer:



•Example.

Problem: Find the inverse of the function
shown.

{(0, 0), (1, 1), (2, 4), (3, 9), (4, 16)}

Answer:

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