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

. and Find gfgf

574)( and 325)(Let 22 xxxgxxxf

Solve the following:

57)( and 5.2127)(Let 1. 22 xxgxxxf

57)( and 5.2127)(Let 2. 22 xxgxxxf

. and find following theofeach For gfgf

xxgxxf 3)( and 720)(Let 3.

234)( and 2)(Let 4. xxgxxf

22 6)( and 3)(Let 5. xxgxxxf

Example

13)( and 5)(Let 2 xxgxxf

.g

f and Find gf State any domain restrictions.

Example 4

. and find following theofeach For g

fgf

25)( and 13)(Let 1. 2 xxgxxf

xxgxxf 3)( and )(Let 2. 2

62)( and 17)(Let 3. xxgxxf

xxgxxxf 5)( and 3)(Let 4. 2

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

xxgxxf 3)( and 1)(Let 2

.fg and Find gf

2)( and 23)(Let xxgxxf

Example 2

.fg and find following theofeach For gf

xxgxxf 2)( and 1)(Let 1.

32)( and 3)(Let 2. xxgxxf

4)( and 32)(Let 3. xxgxxf

32)( and 2)(Let 4. 2 xxgxxf

3)( and 1-3x4)(Let 5. 2 xgxxf

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.

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

Honors Algebra II- Section 2.4 Level B

End Section 2.4