Question

given f(x) = x^2 and g(x)= squar root x+4. determine the following compostions and their doamin and range
a) f(g(x))
b) g-1(superscript) (f(x))
c) g(f(x)) +g-1(superscript)(f(x))

Answer 1

f(x) = x^2
g(x) = √(x+4)

f(g) = g^2 = x+4

g^-1(x) = x^2-4
g^-1(f) = f^2-4 = x^4-4

g(f) = √(f+4) = √(x^2+4)
g(f(x))+g^-1(f(x)) = √(x^2+4)+x^4-4

for domain, just recall that √u is defined only for u>=0

Then determine the range as usual.