1.6 Composite Functions

Introduction

Composing two functions

Composing functions in context

Suppose we know that $y$ is a function of $x$ according to the process defined by $y = f(x) = x^2 - 1$ and, in turn, $x$ is a function of $t$ via $x = g(t) = 3t-4$. Is it possible to combine these processes to generate a new function so that $y$ is a function of $t$?

Let $y = p(x) = 3x - 4$ and $x = q(t) = t^2 - 1$.

Let $r(t) = p(q(t))$. Determine a formula for $r$ that depends only on $t$ and not on $p$ or $q$.

$p(q(0))$

$q(p(0))$

Let functions $p$ and $q$ be given by the graphs in the figure below, which are each piecewise linear - that is, parts that look like straight lines are straight lines. In addition, let $f$ and $g$ be given by the tables below.

$x$ | 0 | 1 | 2 | 3 | 4

$f(x)$ | 6 | 4 | 3 | 4 | 6

$g(x)$ | 1 | 3 | 0 | 4 | 2

Compute each of the following quantities or explain why they are not defined.

$(f \circ g)(2)$

$(g \circ f)(3)$

$g(f(0))$

For what value(s) of $x$ is $f(g(x)) = 4$?

For what value(s) of $x$ is $q(p(x)) = 1$?

Determine a formula for the new function $H = (G \circ D)$ that depends only on the variable $N$.

What is the meaning of the function you found in (a)?

How does a plot of the function $H = (G \circ D)$ compare to that of Dolbear's function? Sketch a plot of $y = H(N) = (G \circ D)(N)$ on the blank axes to the right of the plot of Dolbear's function, and discuss the similarities and differences between them. Be sure to label the vertical scale on your axes.

What is the domain of the function $H = G \circ D$? What is its range?