f(x)=10/4-x^2

Question

find the range of the following

Answer 1

f(x)=10/(4-x²) has vertical asymptotes at x=-2 and x=2. The maximum value of 4-x² is 4 (when x=0) so f(x)=10/4=5/2=2.5, making (0,2.5) a minimum. For -2<x<2, as x→2 or x→-2, 4-x²→0 so f(x)→∞.

When the magnitude of x is large (positive or negative) f(x)→0 (the x-axis), the horizontal asymptote.

When x<-2 or x>2, f(x)<0, so the horizontal asymptote is approached from the negative side. As x approaches -2 or 2, f(x)→-∞.

Therefore the range is {[2.5,∞) (-∞,0)}.

The graph of f(x) has 3 parts: x<-2, -2<x<2, x>2 these parts being separated by asymptotes.