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Chapter 1: Functions and Graphs. Section 1-4: Building Functions from Functions. Objectives. You will learn about: Combining functions algebraically Composition of functions Relations and implicitly defined functions Why:

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chapter 1 functions and graphs

Chapter 1:Functions and Graphs

Section 1-4:

Building Functions from Functions

objectives
Objectives
  • You will learn about:
    • Combining functions algebraically
    • Composition of functions
    • Relations and implicitly defined functions
  • Why:
    • Most of the functions that you will encounter in calculus and in real life can be created by combining or modifying other functions
vocabulary
Vocabulary
  • Composition f of g
  • Relation
  • Implicitly
sum difference product and quotient
Sum, Difference, Product and Quotient
  • Let f and g be two functions with intersecting domains. Then, for all values of x in the intersection, the algebraic combinations of f and g are defined by the following rules:
    • Sum:_________________
    • Difference:_____________
    • Product:_______________
    • Quotient:______________
composition of functions
Composition of Functions
  • Let f and g be two functions such that the domain of f intersects the range of g. The composition f of g, denoted (f ᵒ g)(x) is defined by the rule:
  • (f ᵒ g)(x) = f(g(x))
example 2 composing functions
Example 2Composing Functions
  • Let:

Find (f ᵒ g)(x) and (g ᵒ f)(x) and verify that they are not the same.

example 3 finding the domain of a composition
Example 3Finding the Domain of a Composition

Find the domains of the composite functions:

(g ᵒ f)(x) and (f ᵒ g)(x).

example 4 decomposition of functions
Example 4Decomposition of Functions
  • For each function h, find the functions f and g such that h(x) = f(g(x)).
example 5 modeling with function composition
Example 5Modeling with Function Composition
  • In the medical procedure known as angioplasty, doctors insert a catheter into a heart vein (through a peripheral vein) and inflate a small, spherical balloon on the tip of the catheter. Suppose the balloon is inflated at a constant rate of 44 cubic millimeters per second.
    • Find the volume after t seconds.
    • When the volume is V, what is the radius r?
    • Write an equation that gives the radius r as a function of time. What is the radius after 5 s?
relation
Relation
  • The general term for a set of ordered pairs (x, y) is a relation.
  • If the relation happens to relate a single value of y to each value of x, then the relation is also a function.
example 6 verifying pairs in a relation
Example 6Verifying Pairs in a Relation
  • Determine which ordered pairs (2, -5), (1, 3), and (2, 1) are in the relation defined by x2y + y2 = 5.
  • Is the relation a function?
implicitly
Implicitly
  • Recall the circle x2 + y2 = 4.
  • This relation itself is not a function, but it can be split into two equations that do define functions.
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