*This preview shows
page 1. Sign up
to
view the full content.*

**Unformatted text preview: **tack deconstructing G from an operational approach. Given an input x, the ﬁrst step
is to square x, then add 1, then divide the result into 2. We will assign each of these steps a
function so as to write G as a composite of three functions: f , g and h. Our ﬁrst function,
f , is the function that squares its input, f (x) = x2 . The next function is the function that
adds 1 to its input, g (x) = x + 1. Our last function takes its input and divides it into 2,
2
h(x) = x . The claim is that G = h ◦ g ◦ f . We ﬁnd (h ◦ g ◦ f )(x) = h(g (f (x))) = h(g x2 ) =
2
2+1 =
hx
= G(x).
x2 +1
√ 3. If we look H (x) = √x+1 with an eye towards building a complicated function from simpler
x− 1
√
functions, we see the expression x is a simple piece of the larger function. If we deﬁne
√
f (x)+1
f (x) = x, we have H (x) = f (x)−1 . If we want to decompose H = g ◦ f , then we can glean
the formula from g (x) by looking at what is being done to f (x). We ﬁnd g (x) = x+1 . We
x...

View
Full
Document