Composition of Functions & Difference Quotient

Module 2, Part 2

Composition of Functions and DomainIf and g are functions, then the composite function, or composition, of g and is defined by ( ) .x xg f g f

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

g f

Real World Connection…

Suppose that an oil tanker is leaking oil and we want to be able to determine the area of the circular oil patch around the ship. It is determined that the oil is leaking from the tanker in such a way that the radius of the circular oil patch around the ship is increasing at a rate of 3 feet per minute.

Suppose that an oil tanker is leaking oil and we want to be able to determine the area of the circular oil patch around the ship. It is determined that the oil is leaking from the tanker in such a way that the radius of the circular oil patch around the ship is increasing at a rate of 3 feet per minute.

r(t) = the radius r of the oil patch at any time t

A(r) = the area of a circle as a function of the radius r.

What if we wanted to determine the the area of the oil patch at any given time?

We would need to determine the composition of the function, or A(r (t)). In simple terms, you substitute the r(t) function into the A(r) function.

Solution to Real World Connection…

Difference Quotient()() fxhfx

h

Why the Difference Quotient?

For a function f, the formula 

• This formula computes the slope of the secant

line through two points on the graph of f. These are the points with x-coordinates,  x and x + h.

• The difference quotient is used in the definition

the derivative, a key foundation of Calculus.

()() fxhfxh

FINDING THE DIFFERENCE QUOTIENT

Let (x) = 2x2 – 3x. Find the difference quotient and simplify the expression.

Solution

Step 1 Find the first term in the numerator, (x + h). Replace the x in (x) with x + h.

2( ) 2( ) 3( )x h x h x h f

Step 2 Find the entire numerator ( ) ( ).x h x f f

2 2( ) ( ) 2( ) 3( ) (2 3 )x h x x h x h x x f fSubstitute from previous slide

2 2 22( 2 ) 3( ) (2 3 )x xh h x h x x Remember this

term when squaring x + h

Square x + h

Step 2 Find the entire numerator ( ) ( ).x h x f f

2 2 22( 2 ) 3( ) (2 3 )x xh h x h x x

2 2 22 4 2 3 3 2 3x xh h x h x x

Distributive property

24 2 3xh h h

Combine terms.

Step 3 Find the quotient by dividing by h.

Substitute.2( ) 2( ) 4 3x h x

hh

hxh h f f

(4 2 3)h x hh

Factor out h.

4 2 3x h Divide.

Caution Note: (x + h) is not the same as (x) + (h).

For (x) = 2x2 – 3x in the previous ex…2

2 2

( ) 2( ) 3( )

2 24 3 3

x h x h x h

x h xh hx

f

but 2 2

2 2

( ) ( ) (2 3 ) (2 3 )

2 3 2 3

x h x x h h

x x h h

f f

**These expressions differ by 4xh.

Go to Module #2 Assignment

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