What is the range of the function f(x) all real numbers all real numbers less than

(1)  R = All real numbers  : R   = (- ∞ , ∞)  

(2)  R = All real numbers less than or equal to 3  :   R  = (- ∞ , 3]  

(3)  R  = All real numbers less than or equal to -3    : R  = (- ∞ , -3]  

(4) R = All real numbers greater than or equal to -3  :  R  = [-3 , ∞)

Step-by-step explanation:

RANGE : The set of all output values of a function  f(x) is called the Range Set of the given function.

Now, here the given function is f(x). Here,

(1)  R = All real numbers

So, the range of function f(x)  = (- ∞ , ∞)  = R

(2)  R = All real numbers less than or equal to 3

 So, the range of function f(x)  is set of all x, such that  x ≤ 3

 or R  = (- ∞ , 3]  

(3)  R  = All real numbers less than or equal to -3

So, the range of function f(x)  is set of all x, such that  x ≤ -3

 or R  = (- ∞ , -3]  

(4) R = All real numbers greater than or equal to -3

So, the range of function f(x)  is set of all x, such that  x ≥ -3

 or R  = [-3 , ∞)

ln(2)

Step-by-step explanation:

In order to minimize ln[(x – 3)^2 + 2], we have to minimize [(x – 3)^2 + 2] since natural log is an increasing function

For that, it suffices to minimize (x – 3)^2, which is non-negative for all real numbers x

As such, the minimum value is achieved when x - 3 = 0; - 5 < 3 < 5

Substitute x = 3 in the original equation to get ln(2)