Combining functions Combining functions algebraically, composite algebraically, composite

functions, and functions, and decomposing functions! decomposing functions!

Onward to Section Onward to Section 1.4a…1.4a…

Definition: Sum, Difference, Product, and Quotient ofFunctions

Let f and g be two functions with intersecting domains. Then forall values of x in the intersection, the algebraic combinations off and g are defined by the following rules:

Sum: f g x f x g x

In each case, the domain of the new function consists of allnumbers that belong to both the domain of f and the domain of g.As noted, the zeros of the denominator are excluded from thedomain of the quotient.

Difference: f g x f x g x Product: fg x f x g x

Quotient: ,

f xfx

g g x

0g x provided

Guided PracticeGuided PracticeFor the given functions, find f + g, f – g, fg, f/g, and gg. Givethe domain of each.

2f x x 1g x x

First, find the domain of the original functions, and determinewhere these two domains intersect (overlap).

Domain of f: , Domain of g: 1,

Domain intersection: 1,

This intersection becomes the domain of all of the algebraiccombination functions!!!

Guided PracticeGuided PracticeFor the given functions, find f + g, f – g, fg, f/g, and gg. Givethe domain of each.

with D:

f g x f x g x 2 1x x 1,

f g x f x g x 2 1x x

fg x f x g x 2 1x x

2f x x 1g x x

with D: 1,

with D: 1,

Guided PracticeGuided PracticeFor the given functions, find f + g, f – g, fg, f/g, and gg. Givethe domain of each.

with D: f xf

xg g x

2

1

x

x

1,

with D: gg x g x g x 21x 1,

Can we simplify this last one???Can we simplify this last one???

2f x x 1g x x

Guided PracticeGuided PracticeFor the given functions, find formulas for the functions f + g,f – g, and fg. Give the domain of all functions.

Domain of all 3 combination functions:

5f x x 3g x x

D : 5, D : ,

5 3f g x x x

5 3f g x x x

3 5fg x x x

5,

Guided PracticeGuided PracticeFor the given functions, find formulas for f/g and g/f. Give thedomain of all functions.

2f x x 4g x x

2

4

f xx

g x

D : 2, D : 4,

2

4

x

x

D : 2,

4

2

g xx

f x

4

2

x

x

D : 2,

Now on to Now on to composite composite functions?!functions?!

Definition:Definition:Composition of FunctionsComposition of Functions

Let f and g be two functions such that the domain of f intersectsthe range of g. The composition of f and g, denoted f g, isdefined by the rule

The domain of f g consists of all x-values in the domain of gthat map to g(x)-values in the domain of f.

NOTE: In most cases, f g and g f are different functions!!!

f g x f g x

A Few Practice Problems…A Few Practice Problems…

xf x e

For the given functions, find (f g)(x) and (g f)(x) and verify(both algebraically and graphically) that the two compositefunctions are not the same.

g x x

f g x f g x f x xe g f x g f x xg e xe

Now, how do we verify???Now, how do we verify???

A Few Practice Problems…A Few Practice Problems…

2 1f x x

For the given functions, find (f g)(x) and (g f)(x) and give thedomain of each composition function.

g x x

f g x 2 1x

g f x 2 1x

0,D:

, 1 1, D:

Let’s check these with the calculator…Let’s check these with the calculator…

A Few Practice Problems…A Few Practice Problems…

For the given functions, find (f g)(3) and (g f)(–2).

2 1f x x 2 3g x x

3 3f g f g

8

2 2g f g f

3

Decomposing Functions…working Decomposing Functions…working backwards or undoing a composition…backwards or undoing a composition…

21 3 1 4h x x x

For each function h, find functions f and g such that h(x) = f(g(x)).

2 3 4f x x x 1g x x

Decomposing FunctionsDecomposing Functions

3 1h x x

For each function h, find functions f and g such that h(x) = f(g(x)).

f x x 3 1g x x

Any other ways to solve this one?!?!Any other ways to solve this one?!?!

Let’s do some Let’s do some modeling…modeling…

In the medical procedure known as angioplasty, doctors insert acatheter into a heart vein and inflate a small, spherical balloonon the tip of the catheter. Suppose the balloon is inflated at aconstant rate of 44 cubic millimeters per second.

In math-land, not fashion-land…In math-land, not fashion-land…

1. Find the volume after t seconds.V = 44V = 44tt

Let’s do some Let’s do some modeling…modeling…

34π3r V

In the medical procedure known as angioplasty, doctors insert acatheter into a heart vein and inflate a small, spherical balloonon the tip of the catheter. Suppose the balloon is inflated at aconstant rate of 44 cubic millimeters per second.

In math-land, not fashion-land…In math-land, not fashion-land…

2. When the volume is V, what is the radius r ?

3 3

4π

Vr

33

4π

Vr

Let’s do some Let’s do some modeling…modeling…

3 33 44 33

4π π

t tr

In the medical procedure known as angioplasty, doctors insert acatheter into a heart vein and inflate a small, spherical balloonon the tip of the catheter. Suppose the balloon is inflated at aconstant rate of 44 cubic millimeters per second.

In math-land, not fashion-land…In math-land, not fashion-land…

3. Write an equation that gives the radius r as a function of the time. What is the radius after 5 seconds?

44V t 33

4π

Vr and

At 5 seconds,At 5 seconds,r = 3.745 mmr = 3.745 mm

Whiteboard PracticeWhiteboard PracticeFor the given functions, find formulas for the functions f + g,f – g, and fg. Give the domain of all functions.

21f x x 3g x x

f g x 2 3 4x x

f g x 2 2x x

( )fg x 3 25 7 3x x x

Domain of all five functions: ,

Find f(g(x)) and g(f(x)). State the domain of each.

Whiteboard Whiteboard PracticePractice

2( ) 1f x x 1

( )1

g xx

2

1( ( )) 1

( 1)

: ( ,1) (1, )

f g xx

D

2

1( ( ))

2

: ( , 2) ( 2, 2) ( 2, )

g f xx

D

Find f(x) and g(x) so that the function can be described as y=f(g(x)).

one possible solution…Homework: p. 127-128 1-23 odd

Whiteboard Whiteboard PracticePractice

2

3

( ) ( 1)

( )

f x x

g x x

3 2( 1)y x