1. ( f(x) = x + 2 )
2. ( f(x) = \sqrt{x^2 + 1} )

Step-by-Step Explanation:

1. Function 1: ( f(x) = x + 2 )

   • This is a linear function. It adds 2 to the input ( x ).
   • For any input ( x ), the output is the input plus 2.
   • Example:
     • If ( x = 3 ), then ( f(3) = 3 + 2 = 5 ).
     • If ( x = -1 ), then ( f(-1) = -1 + 2 = 1 ).

2. Function 2: ( f(x) = \sqrt{x^2 + 1} )

   • This function computes the square root of ( x^2 + 1 ).
   • It first squares the input ( x ), adds 1 to the result, and then takes the square root of that sum.
   • Key Points:
     • ( x^2 ) is always non-negative, so ( x^2 + 1 ) is always positive, ensuring the square root is defined.
     • This function grows faster than the linear function because it includes a square root, which increases the output more rapidly as ( x ) increases.
   • Example:
     • If ( x = 3 ), then ( f(3) = \sqrt{3^2 + 1} = \sqrt{1} \approx 3.16 ).
     • If ( x = -2 ), then ( f(-2) = \sqrt{(-2)^2 + 1} = \sqrt{5} \approx 2.24 ).

Summary:

• The first function is straightforward, adding 2 to the input.
• The second function is a bit more complex, involving both squaring and square rooting, which affects the output in a non-linear way.

This difference in structure and the operations involved leads to different output behaviors for each function.