Combining Functions Lesson 5 1 Functions to Combine

Combining Functions Lesson 5. 1

Functions to Combine l Enter these functions into your calculator

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

Combining Functions l l l Turn off the two original functions (F 4) Use them in the expression for the combined function How does this differ from a parabola?

Application l Given two functions having to do with population l l l 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 l Window at 0 < x < 100 and 0 < y < 1000

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

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

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

Combinations Using Tables l Determine the requested combinations x -2 -1 0 1 2 3 r(x) 5 5 6 7 8 9 s(x) -2 2 s(x)/r(x)-s(x) 4 – 2 r(x)

Assignment A l l l Lesson 5. 1 A Page 346 Exercises 1 – 25 odd, 61, 62

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

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

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

Try It Out l q ( p(x) ) = l l q ( 2 x + 1) = (2 x + 1)2 – 3 = 4 x 2 + 4 x + 1 – 3 = 4 x 2 + 4 x - 2

Using the Calculator l Given l Define these functions on your calculator

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

Using the Calculator l Is it also possible to have a composition of the same function? l 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 l 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 l Results of k( j(x) )

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

Assignment B l l l Lesson 5. 1 B Page 347 Exercises 27 – 77 EOO