Question

Let f(x)= x^2 + 5. Find formulas for f(f(x)) and f(f(f(x)))

Answer 1

f(f(x)) means subsitute f(x) or x^2+5 for x in the other one
f(f(x))=(f(x))^2+5
f(x^2+5)=(x^2+5)^2+5
f(f(x))=x^4+10x^2+25+5
f(f(x))=x^4+10x^2+30

f(f(f(x)))
just subsitutte f(f(x)) for x in f(x)
f(f(f(x)))=(x^4+10x^2+30)^2+5
f(f(f(x)))=(x^4+10x^2+30)^2+5
f(f(f(x)))=x^8+20x^6+150x^4+500x^2+625+5
f(f(f(x)))=x^8+20x^6+150x^4+500x^2+630


so


f(f(x))=x^4+10x^2+30
f(f(f(x)))=x^8+20x^6+150x^4+500x^2+630