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Question

Let f:RR be a function such that for any irrational number r, and any real number x we have f(x)=f(x+r). Then, f is

A
an identity function
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B
a constant function
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C
a zero function
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D
onto function
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Solution

The correct option is A 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(x)=f(x+r)
it means it is a constant function,
example:
f(x)=5=5(x)0
f(x+r)=5(x+r)0
f(x+r)=5×1
f(x+r)=5=f(x)

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