Solve this: Q EXAMPLES 8 If ƒ : R →R be defined as follows : - Maths - Relations and Functions

Question

Solve this:

Q EXAMPLES 8 If ƒ : R → R be defined as
follows :

ƒ (x) = 1 , i f x ∈ Q - 1 , i f x ∉ Q

Find (i) ƒ (1/2), ƒ ( π ), ƒ 2 (ii) Range
oƒƒ (iii) pre- images oƒ 1 and – 1

Shivam Mishra · Accepted Answer

Dear Student,
Please find below the solution to the asked query:

We havefx=1,ifx∈Q-1,ifx∉Qi e
fx=1,ifx∈Rational Number-1,ifx∉Irrational NumberAs 12 is rational number, hencef12=1As π and 2 are irrational number hencefπ=-1f2=-1As clear from function definition, range of function is -1,1As functional value of 1 is achieved for rational numbers and -1 is acheived for irrational numbers, hencepre-image of 1 is all rational numbers and pre-image of -1 is all irrational numbers

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Solve this: Q EXAMPLES 8. If ƒ : R → R be defined as follows : ƒ (x) = 1 , i f x ∈ Q - 1 , i f x ∉ Q Find (i) ƒ (1/2), ƒ ( π ), ƒ 2 (ii) Range oƒƒ (iii) pre- images oƒ 1 and – 1.

Solve this:

Q EXAMPLES 8. If ƒ : R $\to$ R be defined as follows :

ƒ (x) = $\{\begin{matrix} 1, & i f & x \in Q \\ - 1, & i f & x \notin Q \end{matrix}$

Find (i) ƒ (1/2), ƒ ( $π$ ), ƒ $(\sqrt{2})$ (ii) Range oƒƒ (iii) pre- images oƒ 1 and – 1.