Composite unctions [Type the document subtitle] Composite unctions, workin them out. luxvis 11/19/01
Composite Functions What are they? In the real world, it is not uncommon or the output o one thin to depend on the input o another unction. For example the amount o tax we would pay depends on the ross salary the person makes. Such unctions are called composite unctions. So a unction is perormed irst and then a second unction is perormed on the result o the irst unction, that is what is actually takin place when we composition. Special terminoloy The composite unction, the composition o and is deined as ollows ( )( x) ( ( x)) For the above unction to be deined or to exist then a certain condition must be met namely Rane ( x) domain ( x) Formula Deinitions What it means ( )( x) ( ( x)) For to be deined Rane ( x) domain ( x) What is the domain o domain domain What is the rane o rane rane Example- ind Let us consider an example and we will see how this works in practice Consider the ollowin two unctions ( x) x and ( x) x 4 Say we will like to ind the ollowin composite unction and whether this unction exists Step 1- sketch both unctions to see what they look like and determine their domains and ranes Sketch the two unctions and ind their respective domains and ranes ( x) x Equation o the raph ( x) x Domain o the raph- what is the values o x it can have, Rane o the raph- values o y it can have [0, )
Now to sketch the raph o ( x) x 4 Equation o the raph ( x) x 4 Domain o the raph- what is the values o x it can have,, Rane o the raph- values o y it can have Now the reason we plotted these two raphs is to help us understand the restrictions that must be placed on the domain or the various composite unctions to be deined Step-- Work out Find ( )( x) ( ( x)) And this ives us the ollowin ( x) ( ( x)) ( x 4) ( x 4) Step -3- Work out i the unction is deined Now or this unction to be deined the condition Rane ( x) domain ( x) This means that the rane is within the domain o the ( x ) Let s see how this looks We have the ollowin operation x ( x) ( x) Domain (x) Rane (x) Domain : R - (-, ) Rane: R - (-, ) Domain (x) Rane (x) Domain: R - (-, ) Rane: R- (0, )
This is a way o showin the step visually. Notice how I am usin the number line to my advantae! So since rane o x ( ) is a subset o the domain o ( x ) then this composite unction exists. I like to put the domain o ( x ) on the bottom and the rane o x ( ) on the top, as it helps me see i the rane is within the domain o ( x ) here are the answers What it means ( )( x) ( ( x)) x 4 For to be deined Rane ( x) domain ( x) What is the domain o domain domain What is the rane o rane rane which is, which is, Example- Now let s see i we can ind the composite unction Let s ollow the previous steps ( x) ( ( x)) x ( ) x 4 For this to be deined then Rane ( x) domain ( x) Now this is the most important step as it shows the actual process that is takin place here
x ( x) ( x) So that ives us the ollowin Domain (x) Rane (x) [0, ) Domain (x) (-, ) Rane (x) Now remember that Rane ( x) domain ( x) The rane o (x) is a subset o the domain o (x), so this composite unction is deined. Important Points to Remember Formula- Deinitions What it means ( )( x) ( ( x)) For to be deined Rane ( x) domain ( x) What is the domain o domain domain What is the rane o rane rane
Diicult Example Consider the ollowin two unctions :{ x : x 3} R, ( x) 3 x and a) Show that is not deined R R x x :, ( ) 1 Let us sketch both raphs and work out their domains and ranes beore we answer the question :{ x : x 3} R, ( x) 3 x The raph that is been plotted :{ x : x 3} R, ( x) 3 x Domain o the raph,3 Rane o the raph 0, The raph that is been plotted : R R, ( x) x 1 Domain o the raph, Rane o the raph 1,
Now or to be deined the condition : R R, ( x) x 1 Let s see how this looks We have the ollowin operation x ( x) ( x) Domain (x) Rane (x) Domain (x) Rane (x) So it is clear that rane (x) is not a subset o the domain (x). So the composite unction is not deined. O course we could deine it i we restrict the domain o to [-1, 3]. Remember domain domain
Composite Functions What does ( x) 1 mean? How do we know i exists? It means we substitute unction (x) into (x) For it to exist or be deined then it must meet this condition: : R R, ( x) x 1 :{ x : x 3} R, ( x) 3 x Let say (x) = 3x and (x) = x+ Find Answer 3 What is the best way o doin these type o questions One way is to ind the domain or each unction and then plot the rane o (X) and the domain o (x) and see i the condition meets :{ x : x 3} R, ( x) 3 x = (x+) = 3(x+) = 3x+6 Now is this deined? : R R, ( x) x 1 First we ind domain and rane o each o the above unction rane o (x) = R = (-, ) domain o (x) = R = (-, ) Draw a number line and you will see that rane is a subset or equal to the domain o (x) Rane (x): (-, Domain (x) :(-, 4 5 What happens i it is ( x ) What is the deinition or ( x ) Everythin is done the same except this time the deinition For domain domain rane rane For domain domain rane rane