Question

Next consider what happens when two of these functions are composed. Suppose g identifies ng regions of (0, 1)d onto (0, 1)d and f identifies nf regions of (0, 1)d onto (0, 1)d . How many regions of its input does f ◦ g(·) identify onto (0, 1)d ?

Answer 1

$n^{2}fg$ is regions of f ◦ g(·).

Step-by-step explanation:

When you multiply two functions together, you'll get a third function as the result, and that third function will be the product of the two original functions.

For example, if you multiply f(x) and g(x), their product will be h(x)=f.g(x), or h(x)=f(x)g(x).

Here we have two functions, f identifies n f regions of (0, 1)d onto (0, 1)d which is equivalent to f(x) = n f. And, g identifies n g regions of (0, 1)d onto (0, 1)d which is equivalent to g(x)= n g. Now,

⇒ ( f × g ) (x ) = f(x) × g(x)

⇒$( fg )(x) = f(x).g(x)\\( fg )(x) = nf.ng\\(fg)(x) = n^{2}fg$

Therefore, $n^{2}fg$ is regions of f ◦ g(·).