Let f: R →R be defined by f(x)={ 1, if x is rational 0, if x is irrational . then ff

step 1 In this question the function is defined in two parts. It is told that you have to take your fx

Let f: R →R be defined by f(x)={ 1, if x is rational ...

Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be defined by $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{l}1, \text { if } \mathrm{x} \text { is rational } \\ 0, \text { if } \mathrm{x} \text { is irrational }\end{array}\right.$ then $\mathrm{ff}(\mathrm{x})$ is (a) $x$ (b) $\mathrm{f}(\mathrm{x})$ (c) 0 (d) 1

Key Concepts

Range Confinement and Its Effect in Composition

When a function's output is limited to a specific subset of values, this constraint can simplify the evaluation of its composition. In the context of f(f(x)), recognizing that f(x) only takes on certain values (e.g., 0 and 1) allows for a more straightforward analysis of the composite function.

Properties of Rational Numbers in Function Definitions

Understanding the characteristics of rational numbers is important when they are used to define a function. Since the definition may treat rational and irrational numbers differently, knowing that specific values like 0 and 1 are rational is critical, as it affects how the function behaves upon composition.

Function Composition

Function composition involves applying one function to the result of another. In particular, when evaluating a composite function like f(f(x)), you first determine the output of f(x) and then apply the function f to that result. This concept is central to understanding how complex functions are built from simpler ones in many areas of mathematics.

Piecewise Functions

Piecewise functions are defined by using different expressions for different parts of their domain. They require evaluating which condition or case the input satisfies, and then using the corresponding rule. This approach is essential when dealing with functions that behave differently under various conditions.

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