(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)