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Chapter 1: Functions and GraphsPowerPoint Presentation

Chapter 1: Functions and Graphs

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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

- Composition f of g
- Relation
- Implicitly

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:______________

Example 1Defining New Functions Algebraically

- Let:

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 2Composing Functions

- Let:

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

Example 3Finding the Domain of a Composition

Find the domains of the composite functions:

(g ᵒ f)(x) and (f ᵒ g)(x).

Example 4Decomposition of Functions

- For each function h, find the functions f and g such that h(x) = f(g(x)).

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

- 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 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

- 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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