if f(x) = x² - and g(x) = 2x - 3, what is the range of (f o g)(x)?
(-∞,∞)
[-1, ∞)
[-5, ∞)
[-4, ∞)

Question

if f(x) = x² -  and g(x) = 2x - 3, what is the range of (f o g)(x)?
(-∞,∞)
[-1, ∞)
[-5, ∞)
[-4, ∞)

ranga · Answer

Is f(x) = x^2   or  x^2 minus something?

anonymous · Answer

sorry, that was x² - 1

ranga · Answer

f(x) = x^2 - 1     g(x) = 2x - 3
(f o g)(x) = f(g(x)) = (2x - 3)^2 - 1 = 4x^2 - 12x + 9 - 1 = 4x^2 - 12x + 8

That is a vertical, upright parabola.  Put it in the vertex form so you can find the vertex which will be the minimum point.  The range of the parabola will be from the minimum value to +infinity.

ranga · Answer

It would be easy to put it in the vertex form if we start from 
f(g(x)) = (2x - 3)^2 - 1
         = 2(x - 1.5)^2 - 1
Compare this to standard vertex form of a parabola y = a(x - h)^2 + k where (h,k) is the vertex.  We can see h = 1.5 and k = -1.  So the vertex is (1.5, -1).
This is the minimum point of the parabola and the minimum value is -1.
The maximum value is +infinity.

So the range is: [-1, infinity)