Question

Let $f:R\to R$ be a function such that for any irrational number $r,$ and any real number $x$ we have $f\left(x\right)=f(x+r)$. Then, $f$ is

• A. an identity function
• B. a constant function
• C. a zero function
• D. onto function

Answer 1

Answer: (B)

a constant function

A constant function is a function that has the same output value no matter what your input value is. Because of this, a constant function has the form y=k where k is a constant(a single value that doesn't change).

So in here our input is $x$ i f we change our input to say $x+r$

and still output does'nt change i.e.

$f\left(x\right)=f(x+r)$

it means it is a constant function,

example:

$f\left(x\right)=5=5(x{)}^0$

$f(x+r)=5(x+r{)}^0$

$f(x+r)=5\times 1$

$f(x+r)=5=f\left(x\right)$