7.6 – Function Operations. I. Operations with Functions All functions can be multiplied, divided,...

7.6 – 7.6 – Function Function OperationsOperations

I. I. Operations with Operations with FunctionsFunctions• All functions can be multiplied, divided, All functions can be multiplied, divided,

added, and subtractedadded, and subtracted

• *when dividing, remember to put in *when dividing, remember to put in lowest terms via factoring if possible.lowest terms via factoring if possible.

• DomainDomain – any x – values the function – any x – values the function coverscovers

• Example 1: compute the following if Example 1: compute the following if

f(x) = xf(x) = x22 -1 and g(x) = x + 1. -1 and g(x) = x + 1.

– (f * g)(x)(f * g)(x)

– f(x) / g(x)f(x) / g(x)

– (f + g)(x)(f + g)(x)

– (f – g)(x)(f – g)(x)

II. II. Composition of Composition of FunctionsFunctions• When two or more functions are When two or more functions are

combined, you form a composition of combined, you form a composition of functions.functions.

• The output from the first function is used The output from the first function is used as an input into the next function.as an input into the next function.

• Work from the inside out, or “back to Work from the inside out, or “back to front”front”

• For example:For example:

•f(x) = 2x + 3f(x) = 2x + 3

•g(x) = xg(x) = x22 + 1 + 1

•g(f(3))g(f(3))

•First, compute f(3) = 2(3) + 3 = 9First, compute f(3) = 2(3) + 3 = 9

•Second, plug in 9 into g(x) = (9)Second, plug in 9 into g(x) = (9)22 + 1 = 81 + + 1 = 81 + 1 = 821 = 82

• Example 2: given the following Example 2: given the following functions, compute the following.functions, compute the following.

f(x) = x – 2f(x) = x – 2

g(x) = xg(x) = x22

A)A) (f ○ g)(x)(f ○ g)(x)

B)B) (f ○ g)(-5)(f ○ g)(-5)

C)C) (g ○ f)(x)(g ○ f)(x)

D)D) (g ○ f)(11)(g ○ f)(11)

• Example 3: The BMW dealer offers a Example 3: The BMW dealer offers a 10% discount on a 7series. At the 10% discount on a 7series. At the same time, the manufacturer offers a same time, the manufacturer offers a $2000 rebate for each purchaser. If $2000 rebate for each purchaser. If the car cost $37,000.00 is it cheaper the car cost $37,000.00 is it cheaper to apply the discount before or after to apply the discount before or after the coupon?the coupon?

• Example 4: f(x) = 3xExample 4: f(x) = 3x22 + 2x – 8 and + 2x – 8 and

g(x) = x + 2, and state g(x) = x + 2, and state domainsdomains

: -f(x) + 4g(x): -f(x) + 4g(x)

: f(x) * g(x): f(x) * g(x)

: 5f(x) / g(x) : 5f(x) / g(x)

• Example 5: Let f(x) = 4x – 1. FindExample 5: Let f(x) = 4x – 1. Find

[ f(x + h) – f(x) ] / h[ f(x + h) – f(x) ] / h