Composition & Inverse

Basic Concepts

Composition: Use the output of one function as the input for another function.

An inverse function reverses the roles of input and output.

The inverse function does *not* mean 1/f(x)!

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.

$r(t)=?$

Find a formula for the area of the inside of the ripple as a function of its radius.

$A(r)=?$

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.

$A(r(t))=?$

Find a formula for the time after the pebble was dropped into the pond as a function of the radius of the ripple.

Solve $A(r(t))$ for $t(A)$...

In Mathematica

Graph and Table example

Suppose $f$ is defined by the graph, and $g$ is defined by the table:

• Find $f(g(100))$, $g(f(100))$, and $f(f(10))$
• Find $f^{-1}(200)$ and $g^{-1}(200)$

GDP example

The gross domestic product (GDP) of the U.S. is given by $G(t)$, where $t$ is the number of years after 1990, and the units of $G$ are billions of dollars.

What is meant by $G^{-1}(14,441.4)=18$?

The year when the GDP was $14.4... trillion was 18 years after 1990 (that is, 2008).

What is meant by $G(15)=12,638.4$?

Pendulum example

The time that it takes for a pendulum to complete one full back-and-forth swing is called its Period, $T$. The period of a pendulum of length $l$ turns out to be $$T(l)=2\pi\sqrt{\frac{l}{g}}.$$

Find a formula for $f^{-1}(T)=l(T)$ and explain its meaning.