Question

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)?

Answer 1

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 $\R.$