Composite Functions

Composite Functions

Composite Functions

• Sometimes it is useful to breakdown bigger functions into parts

• The most complex function is sometimes just 3 or 4 easier parts

• Let's look at the function,

   $y=3\left(x-5\right)^2+7$  

• What is happening to the input first?

Multiple Choice

What is happening to the input first in the function $y=3\left(x-5\right)^2+7$  ?

• It is multiplied by 3

• It is squared.

• 5 is subtracted from the input

• 7 is added to the input.

YES !! 5 is subtracted from the input !!

• If you got that right, pat yourself on the back

• we can now define the function where 5 is subtracted from an input

• we will call is

   $d\left(x\right)=x-5$  

Multiple Choice

 $y=3\left(x-5\right)^2+7$  

After the 5 is subtracted, what is happening to that number next?

• 5 is subtracted from the number

• It is multiplied by 3.

• It is squared.

• 7 is added to the total

YES !! 2 for 2. Somebody call Channel 5.

• Let's call the second function L

• So

   $L\left(x\right)=x^2$  

• Think of the functions as jobs,  the job of function d is to subtract 5 from things.

• The job of  function L is to square things.

• So now,  let's do them together

Magic Number Time

• Pick a Number, Any Number

• Step 1, Subtract 5 from the number.

• Step 2, Square the result.

• Let's see how smart Mr. Magee is??

• Share your result out loud !!!

Wait a minute, How did he do that, so fast!!!

• Well, let's say you shouted out 81.

• You squared it, so Mr. Magee square rooted it. He has a 9.

• You subtracted 5, so Mr. Magee added 5.

• Mr. Magee is old and has done square roots for a long time.

• Anyone can add 5.

• Let's pretend, Laura shouted out 49. What would we do?

Multiple Choice

So you try it, Sadik shouted out 121, What was his number?

• 11

• 16

• 6

• 14646

Multiple Choice

One more, Anne shouted out 361, What was her number? Hint: She was a big fan of Kobe.

• 24

• 8

• 14

• 130326

d a number then L it

• Think of any number

• Subtract five from the number

• Square it.

• What you did was d it first, then L it.

• Mathematically, it was this

   $L\left(d\left(x\right)\right)$  

• Notice since d is on the inside you would do it first.

• What is  $L\left(d\left(11\right)\right)$  ?

Multiple Choice

What is $L\left(d\left(18\right)\right)$  ?

• 169

• 196

• 225

• 256

Let's try the third step

• In our function

   $y=3\left(x-5\right)^2+7$  

• What is the third step?

• This time just say it out loud or text in chat

• We will call the third step  $o\left(x\right)$  

What are we going to do to the input? (in order)

• We will d it, (subtract 5)

• Then we will L it, (square it)

• Then we will o it, (multiply by 3 or triple it).

• What is the

   $o\left(L\left(d\left(9\right)\right)\right)$  ?

Fill in the Blanks

What is the  $o\left(L\left(d\left(8\right)\right)\right)$  ? 

 $S\left(x\right)$  is easy,  we are just adding 7.

• So think of your last answer.

• Add 7 to it.

• The  $S\left(o\left(L\left(d\left(8\right)\right)\right)\right)=34$  

• Why???

Favorite Number Time Again.

• Pick a number

• Input it into the equation

   $S\left(o\left(L\left(d\left(your\ number\right)\right)\right)\right)$  

• Remember d it first and so on.

• What is your result?

• Do you think Mr. Magee can guess your number?

• What is  $S\left(o\left(L\left(d\left(7\right)\right)\right)\right)$  ?

Fill in the Blanks

What is the  $S\left(o\left(L\left(d\left(10\right)\right)\right)\right)$  ?

Multiple Choice

 $f\left(x\right)=3\left(x-5\right)^2+7$ 
What is  $f\left(10\right)$  ?

• 82

• 75

• 5

• 11

The answers are the same. Why?

• According to the Order or Operations, in

   $f\left(x\right)$  .We subtracted 5,  then we squared it,  then we tripled it,  then we added 7

• Or in other words,  we   d'ed it

• Then we L'ed it

• Then we o'ed it

• Then we S'ed it

Poll

What did you think about this form of Learning Mathematics through Quizizz.com Lessons

• I really liked it.

• I liked it.

• It was something different..

• I did not like it.

• I hated it.

Open Ended

What could Mr. Magee have done better in preparing this lesson?