Essential Question: How do you find the value of f(g(5)) and g(f(3))

given: f(x) = 3x + 1 and g(x) = 2x - 5

Operations with functionsYou can add, subtract, multiply and divide

functions.

Addition(f + g)(x) = f(x) + g(x)

Multiplication (f • g)(x) = f(x) • g(x)

Subtraction (f - g)(x) = f(x) – g(x)

Division( )

( ) , ( ) 0( )

f f xx g x

g g x

Domains The domain of a function is all real numbers that

can be substituted (plugged in) to a function in order to get an answer

The domain for most functions are all real numbers, with two significant exceptions You can’t use a number that would make you divide

by 0. You can’t use numbers that would get you the square

root of a negative number. Only the first rule will apply for us in this chapter.

The domain for the (addition, subtraction, and multiplication) combined functions are all numbers that work for both individual functions.

Example #1 If f(x) = 3x + 8 and g(x) = 2x – 12Find f + g, f – g, and their domains

(f + g)(x) = f(x) + g(x) = (3x + 8) + (2x – 12) = 5x – 4 (f – g)(x) = f(x) – g(x) = (3x + 8) – (2x – 12) = 3x + 8 – 2x + 12 = x + 20

Because the domain for both functions are all real numbers, the domains of f + g and f – g are also all real numbers

Example #2 – YOUR TURN If f(x) = 5x2 – 4x and g(x) = 5x + 1Find f + g, f – g, and their domains

(f + g)(x) = (f – g)(x) =

The domain is:

5x2 + x + 15x2 - 9x - 1

All real numbers

Example #3 If f(x) = x2 - 1 and g(x) = x + 1

Find f • g, , and their domains (f • g)(x) = f(x) • g(x) = (x2 - 1)(x + 1) = x3 + x2 – x – 1 Division on next slide

Because the domain for both functions are all real numbers, the domain of f • g is all real numbers

( )f

xg

Example #3 If f(x) = x2 - 1 and g(x) = x + 1

Find f • g, , and their domains

For this domain, we have to look at the original functions (both all real numbers) as well as the combined.

In this case, we can’t use -1, as that would cause us to divide by 0. So we say the domain is all real numbers except -1 (also written as x ≠ -1)

( )f

xg

2 1( )

1

( 1)( 1)

11

f xx

g x

x x

xx

Example #2 – YOUR TURN If f(x) = 6x2 + 7x - 5 and g(x) = 2x - 1Find f • g, , and their domains

(f • g)(x) = =

The domain is:

12x3 + 8x2 - 17x + 53x + 5

Multiplication: All real numbersDivision: All real numbers except ½

f

g

( )f

xg

AssignmentPage 400 – 4011 – 19, all

Essential Question: How do you find the value of f(g(5)) and g(f(3))

given: f(x) = 3x + 1 and g(x) = 2x - 5

Composite FunctionsA composite function is the result of taking

the results from one function, and then applying them into another.

It uses the symbol o, as like (f o g). Do not confuse the symbol with either the letter “o” or with the symbol for multiplication “•”

(f o g)(x) means f(g(x)), meaning apply the x to the inner-most function first, apply that answer to the next closest function.

Like subtraction, the order the functions are presented in matters.

Example #1, Method 1 If f(x) = x - 2 and g(x) = x2

Find (g o f)(-5) (g o f)(-5) = g(f(-5))

f(-5) = (-5) – 2 = -7 = g(-7) = (-7)2

= 49We apply inside out:

(g o f)(-5)

Example #1, Method 2We can combine the functions as variables

before plugging numbers in If f(x) = x - 2 and g(x) = x2

Find (g o f)(-5) (g o f)(x) = g(f(x)) = g(x – 2) = (x – 2)2

(g o f)(-5) = ((-5) – 2) 2

= (-7)2

= 49

Example #1 – YOUR TURN If f(x) = x - 2 and g(x) = x2

Find (f o g)(-5) and (f o g)() (f o g)(-5) = (f o g)(x) = x2 - 2

23

Example #2 – YOUR TURN If f(x) = x3 and g(x) = x2 + 7Find (g o f)(2) and (f o g)(2)

(g o f)(2) = (f o g)(2) =

711331

Real World Connection, Consumer Issues Suppose you are shopping in the store. The store is

offering 20% off everything in the store. You also have a coupon worth $5 off any item.

a) Use function to model discounting an item by 20% and to model applying the coupon.

b)Use a composition of your two functions to model how much you would pay for an item if the clerk applies the discount first and then the coupon.

c) Use a composition of your two functions to model how much you would pay for an item if the clerk applies the coupon first and then the discount.

d)How much more is any item if the clerk applies the coupon first?

a) 20% discount: f(x) = x – 0.2x = 0.8x$5 coupon: g(x) = x – 5

b) Discount first: (g o f)(x) = g(f(x))= g(0.8x)= 0.8x – 5

c) Coupon first: (f o g)(x) = f(g(x))= f(x - 5)= 0.8(x – 5)= 0.8x – 4

d) Difference: (f o g)(x) - (g o f)(x) = (0.8x – 4) – (0.8x – 5)

= 0.8x – 4 – 0.8x + 5= 1

Meaning: Any item would cost you $1 more if the clerk applied the coupon first

AssignmentPage 400 - 40120 – 43, all

Extra Examples / Quiz Practice – YOUR TURNLet:

o f(x) = 2x + 3o g(x) = x2 – xo h(x) = 3x – 1o j(x) = 2x2

Find the following:o g(x) + h(x) =o 4f(x) – 2h(x) = o (j o g)(3) =o f(1) + g(2) – h(3) • j(4) =

x2 + 2x - 12x + 14

72-249