Question

Find the value of (fof)(1.4327), given f is a step function.

Answer 1

Hint: we are going to use the process of calculating (f o g) = f ((g(x)). Given f is a step function that means a it can be written as a linear combination of indicator function of intervals.

Given that f is a step function.

$ \Rightarrow f(x) = \left[ x \right]$, [ ] is a step function.
$ \Rightarrow (fof)(1.4327) = f\left[ {(1.4327)} \right]$
[since the step function value of 1.4327 is 1]
= $f(1)$
= $\left[ 1 \right]$
= 1 since 1 is a rational number.

Note:
Let’s take two functions f, g. Functions can be added and multiplied just like numbers f + g, f – g, f*g. For the functions f and g we define (f o g) the composition of f and g that is (f o g) = f (g(x)).

We have to apply the g function to x to get g(x) and then apply f to g(x) to get f (g(x)). Here f is the outer function and g is the inner function. But in the given problem we have both inner and outer function is f.