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?

enter image source here

1 Answer
Apr 14, 2018

Please see the explanation below.


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.