Answer :
Final Answer:
The value of ( g(3) ) is 101. Option c) 101 is the correct choice.
Explanation:
1. Find ( f(3) ):
- Given [tex]\( f(x) = x^2 + 1 \),[/tex] plug in ( x = 3 ) to find ( f(3) ).
- [tex]\( f(3) = 3^2 + 1 = 9 + 1 = 10 \).[/tex]
2. Find ( g(3) ):
- Since [tex]\( g(x) = f(f(x)) \),[/tex] we need to find ( f(10) ) to determine ( g(3) ).
- Substitute ( x = 10 ) into the function [tex]\( f(x) = x^2 + 1 \)[/tex] to find ( f(10) ).
- [tex]\( f(10) = 10^2 + 1 = 100 + 1 = 101 \).[/tex]
By following these steps, we determine that ( g(3) ) is indeed 101. This process highlights the importance of understanding function composition and applying it correctly to find the value of a composite function. Through clear calculation and substitution, we derive the correct value of ( g(3) ) as 101, reinforcing the importance of systematic problem-solving techniques in mathematics. Therefore, option c) 101 is the correct choice.
Question
Let ℤ+ denote the set of positive integers. When f(x) = x² + 1, let g be the function defined by g(x) = f(f(x)). What is g(3)?
a) 11
b) 27
c) 101
d) 83
