Domain and Range of Real Functions:

1. f(x) = -|x|:
The function f(x) = -|x| represents a piecewise defined function that takes the input x, finds the absolute value of x, and then multiplies it by -1. To determine the domain and range of this function, we need to consider the possible values for x.

Domain:
The domain of a function refers to the set of all possible input values for which the function is defined. Since the absolute value of any real number is always non-negative, the function f(x) = -|x| is defined for all real numbers.

Domain: All real numbers (-∞, ∞)

Range:
The range of a function refers to the set of all possible output values. In this case, the function takes the absolute value of x and then multiplies it by -1. The absolute value of any number is always non-negative, so the output of the function will always be non-positive.

Range: All non-positive real numbers (-∞, 0]

2. f(x) = √(9 - x^2):
The function f(x) = √(9 - x^2) represents a square root function with a radicand of (9 - x^2). To determine the domain and range of this function, we need to consider the possible values for x.

Domain:
The domain of a square root function is determined by the values that make the radicand (9 - x^2) non-negative. Since the square root of a negative number is undefined in the real number system, we need to find the values of x that make the radicand non-negative.

9 - x^2 ≥ 0

Solving this inequality, we get:
x^2 ≤ 9
-3 ≤ x ≤ 3

Therefore, the function is defined for all values of x between -3 and 3, inclusive.

Domain: [-3, 3]

Range:
The range of a square root function depends on the values of the radicand (9 - x^2). Since the square root of a non-negative number is always non-negative, the range of the function will also be non-negative.

Range: All non-negative real numbers [0, ∞)