What is the domain of the functions f(x)= 1/root(x+|x|) f(x)= x|x|
√(x+|x|) = 0 when x <= 0
so, domain of 1/√(x+|x|) is all reals > 0
domain of x|x| is all reals
To find the domain of a function, we need to determine the set of all possible input values (x) for which the function is defined.
For the function f(x) = 1/√(x + |x|), we need to consider the restrictions on the square root and the absolute value.
1) Square Root Restriction:
The expression inside the square root (x + |x|) cannot be negative, as the square root of a negative number is undefined in the real number system. So, we need to find the values of x that make (x + |x|) greater than or equal to 0.
|x| is always non-negative, meaning it is greater than or equal to 0. Therefore, to determine the domain, we need to find the values of x that make (x + |x|) greater than or equal to 0.
If we split the domain into two cases based on the sign of x:
- For x ≥ 0: (x + |x|) simplifies to (x + x) = 2x
- For x < 0: (x + |x|) simplifies to (x - x) = 0
Considering both cases, we find that the domain of f(x) = 1/√(x + |x|) is all real numbers.
Now, let's consider the next function.
2) f(x) = x|x|
This function includes a multiplication between x and |x|. Since |x| can be either positive or zero, the function remains defined for all real numbers.
Therefore, the domain of f(x) = x|x| is also all real numbers.