FUNCTION COMBINATIONSObjective: SWBAT

~Combine functions using arithmetic and composition

~Determine the domain and range of combined functions

DOMAIN AND RANGEDomain The domain is the set of all x values that work in the function

To find the domain: What kind of numbers can I

plug in for x?? Positive numbers?? Negative numbers?? Zero??

Range The range is the set of all y values that a function outputs

To find the range: You have to think!! What

numbers can you get out of the function??

WHAT ARE THE DOMAINS OF EACH OF THE FOLLOWING?

ARITHMETIC COMBINATIONS OF FUNCTIONSSum:

Difference:

Product:

Quotient:

FIND AND THE DOMAIN OF EACH and

COMPOSITIONS OF FUNCTIONSThe composition of the function with the function is:

The domain of is the set of all in the domain of such that is in the domain of

FUNCTION COMPOSITION RELAY Each person in the group is assigned a function.

The person in the back of your group will evaluate their function for the x value on the next slide and write their answer on an index card.

They will then hand their answer up to the person in front of them, who will evaluate their function for the value handed up.

Lather, rinse, repeat, until the front person gets their answer.

They will raise their hand and pass this answer to me.

First group to pass me the correct answer wins!

FUNCTION COMPOSITION RELAY

Ready? Go!

The first value is .

FUNCTION COMPOSITION RELAYWhat composition did we just find?

Hint: What function did we do first? Second? Third? Fourth?

We always work from the inside out, so the function we started with is on the inside.

FUNCTION COMPOSITION RELAYDoes order matter?

Switch functions with your group members so that for the next round, you will be evaluating Remember, we work from the inside out!

FUNCTION COMPOSITION RELAY

Round two!

Ready? Go!

The second value is

FUNCTION COMPOSITION RELAYSwitch your functions again so that you are evaluating

𝑥=−10

FUNCTION COMPOSITION RELAYLast more time with numbers.

Switch functions to find

𝑥=5

WHAT WAS THE PROBLEM WITH THE LAST ROUND? The number that got handed up for j to be evaluated with wasn’t in the domain of j.

Even though the starting number, 5, was in the domain of each of our functions, 5 is NOT in the domain of the composition .

This means that we need to check to see if the ranges of the inner functions are in the domains of the outer functions.

Luckily, we won’t be doing this with strings of four functions, so it’s not as daunting as it seems!

COMBINATIONS OF FUNCTIONS USING A GRAPHUse the graphs of and to find the following:

COMPOSITION OF FUNCTIONS FROM TABLES

x f(x)

-3 4

0 5

3 1

5 -14

6 -2

Use the tables to evaluate the following:

• f(g(7))

• g(f(-3))

• f(f(0))

• g(f(5))

• f(g(-1))

x g(x)

-1 6

0 -1

4 0

5 8

7 3

COMPOSITION OF FUNCTIONS and . Find f(g(x))

and . Find g(f(x))

DOMAIN OF COMPOSITION FUNCTIONSExcluding values from the domain of

If is not in the domain of then it is not in the domain of Any for which is not in the domain of must not be in the domain of

Always find the domain of by using the domain of and

FINDING COMPOSITIONS OF FUNCTIONS AND THEIR DOMAINS

1. , 2. ,

Find and and then their domains.

IDENTIFYING A COMPOSITE FUNCTIONIn calculus, it is very important to be able to identify two functions that make up a composite function. Basically, this means ‘decomposing’ a composite function and finding an ‘inner’ and ‘outer’ function.

In each of the following, write as a composition of two functions.

1. 2.

CLASSWORK

Complete the back side of your HW worksheet.

Read the directions carefully to make sure you are doing each part

of the problem.

You may work with a partner.

On #14, take the radical off of the function g(x) so it is just

g(x)=2x+4

HOMEWORK

Pg 59 #35-38, 55-60, 63, 67, 72