Combining Functions Lesson 5.1

Functions to Combine Enter these functions into your calculator

Combining Functions Consider the following expressions Predict what will be the result if you graph

Combining Functions Turn off the two original functions (F4) Use them in the expression for the combined function How does this differ from a parabola?

Application Given two functions having to do with population P(x) is the number of people S(x) is the number of people who can be supplied with resources such as food, utilities, etc. Graph these two functions Window at 0 < x < 100 and 0 < y < 1000

Population and Supply Viewing the two functions What is the significance of S(x) – P(x) What does it look like – graph it

Population and Supply What does it mean? When should we be concerned?

Population and Supply Per capita food supply could be a quotient When would we be concerned on this formula? Set window -5 < y < 5

Combinations Using Tables Determine the requested combinations x -2 -1 1 2 3 r(x) 5 6 7 8 9 s(x) s(x)/r(x) r(x)-s(x) 4 – 2r(x)

Assignment A Lesson 5.1A Page 346 Exercises 1 – 25 odd, 61, 62

Composition of Functions Value fed to first function Resulting value fed to second function  End result taken from second function 

Composition of Functions Notation for composition of functions: Alternate notation:

Try It Out Given two functions: Then p ( q(x) ) = p(x) = 2x + 1 q(x) = x2 - 3 Then  p ( q(x) ) = p (x2 - 3) = 2 (x2 - 3) + 1 = 2x2 - 5 Try determining  q ( p(x) ) 

Try It Out q ( p(x) ) = q ( 2x + 1) = (2x + 1)2 – 3 =

Using the Calculator Given Define these functions on your calculator

Using the Calculator Now try the following compositions: g( f(7) ) f( g(3) ) g( f(2) )                f( g(t) ) g( f(s) ) WHY ??

Using the Calculator Is it also possible to have a composition of the same function? g( g(3.5) ) = ???

Composition Using Graphs k(x) defined by the graph j(x) defined by the graph Do the composition of k( j(x) )

Composition Using Graphs It is easier to see what the function is doing if we look at the values of k(x), j(x), and then k( j(x) ) in tables:

Composition Using Graphs Results of k( j(x) )

Composition With Tables Consider the following tables of values:  x 1 2 3 4 7 f(x) g(x) f(g(x) f(g(1)) g(f(x) g(f(3))

Assignment B Lesson 5.1B Page 347 Exercises 27 – 77 EOO