2.4 – Operations with Functions

Objectives: Perform operations with functions

to write new functions Find the composition of two

functions Standard:

2.8.11.S. Analyze properties and relationships of functions

I. Operations With Functions

For all functions f and g:

Sum (f + g)(x) = f(x) + g(x)

Difference (f – g)(x) = f(x) – g(x)

Product (f · g)(x) = f(x) · g(x)

Quotient ( )(x) = , where g(x) ≠ 0gf

)()(xgxf

Example 1

Example 2

gfgf

xxgxxxf

Find b. Find a.

57)( and 5.2127)(Let 22

Example 3

Example 4

ns.restrictiodomain any state and ,g

Find b. Find a.

.25)( and 13)(Let 2

fgf

xxgxxf

Composition of Functions Let f and g be functions of x.

The composition of f and g, denoted f ◦ g, is defined by f(g(x)).

The domain of y = f(g(x)) is the set of domain values of g whose range values are the domain of f. The function f ◦ g is called the composite function of f with g.

Example 1

Example 2Let f(x) = -2x + 3 and g(x) = -2x.

a. Find f ◦ g. b. Find g ◦ f.

Let f(x) = x2 + 4 and g(x) = 2x.a. Find f ◦ g. b. Find g ◦ f.

Example 3

2

b. g(f(x)) = -2 (-2x +3)

= 4x - 6

2

2

a. f(g(x)) = -2 (-2x) + 3

= -8x + 3

2

2

Example 4

Example 5

A local computer store is offering a $40.00 rebate along with a 20% discount. Let x represent the original price of an item in the store. a. Write the function D that represents the sale price

after a 20% discount and the function R that represents the sale price after the $40 rebate.

b. Find the composition functions (R ° D)(x) and (D ° R)(x), and explain what they represent.

a. Since the 20% discount on the original price is the same as paying 80% of the original price, D(x) = 0.8x The rebate function is R(x) = x - 40

b. 20% discount first $40 rebate first

R(D(x)) = R(0.8x) D(R(x)) = D(x – 40)

= (0.8x) – 40 = 0.8(x – 40)

= 0.8x – 40 = 0.8x – 32

Notice that taking the 20% discount first results in a lower sales price.

End Section 2.4

Writing Activities

5. What is the difference between (fg)(x) and (f ◦ g)(x)? Include examples to illustrate your discussion.

6. In general, are (f ◦ g)(x) and (g ◦ f)(x) equivalent functions? Explain.

Homework

Integrated Algebra II- Section 2.4 Level A

Academic Algebra II- Section 2.4 Level B