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Math 170 Functions, Data, and Models

This section covers the fundamental concepts of inverse functions and function composition, essential in mathematics. An inverse function reverses inputs and outputs, while composition uses the output of one function as the input to another. Practical examples include modeling circular ripples created by a pebble dropped in a pond, analyzing pendulum motion, and understanding GDP in economics. We derive formulas for radius, area, and time in relation to these applications, enhancing comprehension of relationships between variables in mathematical models.

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Math 170 Functions, Data, and Models

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  1. Math 170 Functions, Data, and Models 10 Composition and Inverse Section 2.4

  2. Basic Concepts • An inverse function reverses the roles of inputs and outputs. • The composition of two functions consists of using the output of one function as the input to the other function. x y = f(x) f g z = g(y) = g(f(x)) x = f-1(y) y f-1

  3. Pond Example • A rock dropped into a pond creates a circular ripple whose radius increases at a steady rate of 10 cm per second. • Find a formula for the radius of the ripple as a function of the time after the pebble was dropped into the pond. • Find a formula for the area of the inside of the ripple as a function of its radius. • Find a formula for the area of the inside of the ripple as a function of time after the pebble was dropped into the pond. • Find a formula for the time after the pebble was dropped into the pond as a function of the radius of the ripple.

  4. Graph and Table Example • Suppose is defined by the graph and is defined by the table. • Find , , and . • Find and .

  5. Pendulum Example • The period, , of a pendulum of length is given by , where is a constant. • Find a formula for and explain its meaning.

  6. GDP Example • The gross domestic product (GDP) of the US is given by where is the number of years since 1990 and the units of are billions of dollars. • What is meant by ? • What is meant by ?

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