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Why is f not a function from R to R if a) f (x) = 1/x? b) f (x) =√x? c) f (x) = ±√(x^2+1)?

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A function f from \R to \R is a rule that assigns to each element x\in \R exactly one

element, called f(x), f(x)\in \R. in

a) f(x)=\dfrac{1}{x}

x\not=0

The function f is undefined at x=0.

Therefore a function f(x)=\dfrac{1}{x} is not a function from \R to \R.


b) f(x)=\sqrt{x}

x\geq0

The function f is undefined for x\in \R,x<0.

Therefore a function f(x)=\sqrt{x} is not a function from \R to \R.


c) f(x)=\pm\sqrt{x^2+1}

f(0)=\pm\sqrt{0^2+1}=\pm1

A relation f(x)=\pm\sqrt{x^2+1} assigns to x=0 two elements -1 and 1.


Therefore f(x)=\pm\sqrt{x^2+1} is not a function from \R to \R.


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