real valued functions

Question

For the real-valued functions =gx−x^2-5

 and =hx−x-2

, find the composition g∘h

 and specify its domain using interval notation.

Answer 1

g(h(x)) = g(x-2) = (x-2)^2 - 5 = x^2 -4x +4 -5 = x^2 -4x -1

(goh)(x) = x^2 -4x -1

x^2 -4x -1 = (x-2)^2 -5 is an upward opening parabola with vertex or minimum point (2,-5)

the range is [-5, + infinity)

the domain is any real number or in interval notation (-infinity, + infinity)