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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

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Chapter 1 functions and graphs

Chapter 1:Functions and Graphs

Section 1-4:

Building Functions from Functions



  • 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



  • 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:______________

Example 1 defining new functions algebraically

Example 1Defining New Functions Algebraically

  • Let:

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?



  • 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?



  • 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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