Question

What's the range and domain of f(x)=1/(root(x^2+3))? and how to prove it's not one to one function?

Answer 1

Please see the explanation below.

Explanation:

`f(x)=1/sqrt(x^2+3)`
a) The domain of f:
`x^2+3>0` => notice that this is true for all real values of x, thus the domain is:
`(-oo, oo)`

The range of f:
`f(x)=1/sqrt(x^2+3)` => notice that as x approaches to infinity f approaches to zero but never touches y = 0, AKA the x-axis, so the x-axis is a horizontal asymptote. On the other hand the maximum value of f occurs at x = 0, thus the range of the function is:
`(0, 1/sqrt3]`

b) If f: ℝ → ℝ , then f is a one to one function when f(a) = f(b) and
a = b, on the other hand when f(a) = f(b) but a ≠ b, then the function f is not one to one, so in this case:
f(-1) = f(1) = 1/2, but -1 ≠ 1, hence the function f is not one to one on its domain.