Respuesta :
Part A
The reason why the square root of (-x) (√(-x)) is not necessarily undefined,
is because x is a placeholder, such that x can take on a negative value for
which (-x) is a positive number, therefore, √(-x) = √(+ve number) is a
defined real number
Part B
The domain is x ≤ 0
The range is f(x) ≥ 0
The process of arriving at the above statement is as follows;
The given and required information:
Given that √(-144) = Undefined, why is √(-x) not necessarily undefined
The given function, f(x) = √(-x)
Method:
Definition of the square root of a negative number
Definition of a variable x
Find appropriate value for the variable x with a negative value having defined square root
Solution:
Part A
The square root of a real number with a negative value is an imaginary number
Example; √(-144) = 12·√(-1) = 12·i (imaginary number)
The square root of a real number with a positive value or zero, is a real number
√(144) = ±12
A variable, x, is an input value of a function that is a quantity place holder in the function which is a function argument
The given function is f(x) = √(-x)
Where;
x = An argument of the function
When x ≤ 0, such that x is a negative real number or zero, we have;
x = R⁻ or 0
Therefore;
-x= -R⁻ or 0 = R⁺ or 0
-x = R⁺
From which we get;
f(-x)= √(-x) = √(R⁺) = Defined
f(-x) = √(R⁺) ≥ 0 The range of the function is f(x) ≥ 0
An example, when x = -16, we have;
-x = -(-16) = 16
√(-x) √(16) = 4 (Defined)
Part B;
Thee domain where the function is defined is where x = -R⁻, which is x ≤ 0
The output of the function when f(x) is defined which is the range = f(x) ≥ 0
Therefore, the reason why the square root of √(-x) is not necessarily undefined, is because the value of x can represent a negative real number or zero
Learn more about real numbers here;
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