2.4 – operations with functions objectives: perform operations with functions to write new...
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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