Operations on Functions2.7

Folding functions into functions

And keeping track of domains

Start with two functions:

What is the domain for each function?

Combine them through addition:

Combine them through subtraction:

What is (g – f)(x)?

What is the domain of each combined function? Graph and see.

f (x)x2 16 g(x) x

( f g)(x) f (x) g(x)x2 16 x

( f g)(x) f (x) g(x)x2 16 x

Start with two functions:

What is the domain for each function?

Combine them through addition:

Combine them through subtraction:

What is the domain of each combined function? Note how the domain of the combined function is an intersection of the two original domains.

f (x)x2 16 g(x) x

( f g)(x) f (x) g(x)x2 16 x

( f g)(x) f (x) g(x)x2 16 x

x 0

x 0

x 0

Start with two functions:

Now, combine them through multiplication:

Combine them through division:

What is the domain of each combined function? You can anticipate the domains and graph to verify them.

Note that domains of these combined functions are identical for each operation except for division. What is the difference, and why is it there?

f (x)x2 16 g(x) x

( f g)(x) f (x)g(x) x2 16 x

f

g

x f (x)

g(x)x2 16x

Start with two functions:

Now, combine them through multiplication:

Combine them through division:

What is the domain of each combined function?

Note that domains of these combined functions are identical for each operation except for division. What is the difference, and why is it there? In this case, x cannot equal zero at all.

f (x)x2 16 g(x) x

( f g)(x) f (x)g(x) x2 16 x

f

g

x f (x)

g(x)x2 16x

x 0

x 0

x 0

Start with two functions: f (x)x2 16 g(x) x

The tricky operation is when the functions are embedded within each other.

( f og)(x) f (g(x))( x )2 16 x 16

(go f )(x)g( f (x)) x2 16

What are the domains of these functions? Graph…

Start with two functions: f (x)x2 16 g(x) x

The tricky operation is when the functions are embedded in each other.

( f og)(x) f (g(x))( x )2 16 x 16

(go f )(x)(g( f (x)) x2 16

What are the domains of these functions? They can be something altogether different from the original domains.On the 84’s or with CAS you can graph the embedded form.

x 0

x 4

Using the TI-84

Use the sequence of Yn = in your graphing calculator to graph a function as a composite of other functions.

We can consider y = √(x)2 – 16 as f(g(x)) if f(x) = x2 - 16 and g(x) = √(x).

In your calculators type Y1 = √(x) and Y2 = (Y1)2 – 16. Use VARS→Y-VARS to pull up Y1.

Graph Y2 then Y3 = √(x)2 – 16 as a check.

Using the TI-84

Use the sequence of Yn = in your graphing calculator to graph a function as a composite of other functions.

We can consider y = √(x)2 – 16 as f(g(x)) if f(x) = x2 - 16 and g(x) = √(x).

In your calculators type Y1 = √(x) and Y2 = (Y1)2 – 16. Use VARS→Y-VARS to pull up Y1.

Graph Y2 then Y3 = √(x)2 – 16 as a check.

Using CAS

On a graph page, type in one function as, say, f1(x) and the other as f2(x).

We can consider f2(f1(x)) = √(x)2 – 16 as f(g(x)) if f2(x) = (f1(x))2 - 16 and f1(x) = √(x).

No need to pull up variables– just type it in.

Graph f(3) = √(x)2 – 16 as a check.

Try it f (x)x2 16 g(x) x

Find values with these combined functions:

Find ( f g)(4)

(go f )(4)

Try it f (x)x2 16 g(x) x

Find values with these combined functions:

Find ( f g)(4)42 16 4 16 16 2 2

(go f )(4) 42 16 16 16 0

Once again using two different functions

For and find their domains

Find the equations and domains of

What are the values for

f (x) 25 x2 g(x)3x 1

( f g)(4)

( f g)(4)

( f g)(4)

f

g

(4)

( f g)(x)

( f g)(x)

( f g)(x)

f

g

( f og)(x)

(go f )(x)

Once again using two different functions

For and find their domains

Find the equations and domains of

What are the values for

f (x) 25 x2 g(x)3x 1

5,5 5,5

5,5

13

25

1325

1325

1325

2

2

2

x

x

xx

xx

xx

5,3

1

3

1,5

5,5

5,5

5,5

1)25(3

1325

2

2

x

x

5,5

3

4,2

13

3

39133

10133

16133

Look at composite functions using tables

Maybe it will help clarify what's happening in a composite function, as we work from the inside out. We did something similar in that journal.

Find p(2) Find q(p(2)) Find p(q(2)) Find p(q(4))

x 1 2 3 4

q(x) 3 4 2 1

p(x) 4 1 3 2

Working backwards

Express y = (2x + 5)8 as a composite function f(g(x)).

There are an infinite number of combinations that would combine to form this as a composite. Let's find three.

Working backwardsExpress y = (2x + 5)8 as a composite function

f(g(x)).

There are an infinite number of combinations that would combine to form this as a composite. Let's find three.

1. f(x) = x8 g(x) = (2x + 5)2. f(x) = x4 g(x) = (2x + 5)2

3. f(x) = x2 g(x) = ???

Working backwardsEveryone to the board: y = 2(x + 5)3.

Generally speaking, we would want to find the simplest arrangement.

ApplicationsUse a composite function to combine several functions in,

say, a word problem (aack). Read example 6, page 193.

The radius changes at a rate 1.5 cm/sec.

Volume is expressed by V = (4/3)πr3 or V(r) = (4/3)πr3

How would we combine them to express the change in volume as a function of time?

ApplicationsUse a composite function to combine several functions in,

say, a word problem (aack). Read example 6, page 193.

The radius changes at a rate 1.5 cm/sec. So, r = 1.5t or r(t) = 1.5t

V = (4/3)πr3 or V(r) = (4/3)πr3

The change in volume as a function of time:

V(t) = (4/3)π(1.5t)3