Domain and Range of f(x) = √(16 - x²)

To find the domain and range of the function f(x) = √(16 - x²), we need to consider the restrictions on x that will result in a defined value for the function.

Domain:
The domain of a function represents all the possible values that x can take. In this case, we need to determine the values of x that will make the expression under the square root (√) valid.

The expression inside the square root, 16 - x², must be non-negative (≥ 0) since we cannot take the square root of a negative number. Therefore, we solve the inequality:

16 - x² ≥ 0

This inequality can be rewritten as:

x² - 16 ≤ 0

Now, we factor the inequality:

(x - 4)(x + 4) ≤ 0

The critical points occur when (x - 4)(x + 4) = 0. Solving this equation, we find x = -4 and x = 4.

Key Points:
- The critical points of the inequality x² - 16 ≤ 0 are x = -4 and x = 4.

Now, we create a sign chart to determine the sign of (x - 4)(x + 4) in different intervals:

Interval (x - 4) (x + 4) (x - 4)(x + 4)
(-∞, -4) - - +
(-4, 4) - + -
(4, ∞) + + +

Key Points:
- From the sign chart, we observe that (x - 4)(x + 4) is negative in the interval (-4, 4).

Therefore, the domain of the function f(x) = √(16 - x²) is the set of all x-values that make the inequality (x - 4)(x + 4) ≤ 0 true. This can be written as:

Domain: -4 ≤ x ≤ 4

Range:
The range of a function represents all the possible values that the function can take. In this case, the range of the function f(x) = √(16 - x²) is determined by the output values of the square root function.

Key Point:
- The square root function (√) always returns a non-negative value (≥ 0).

Since the expression inside the square root is always non-negative (≥ 0), the range of the function f(x) = √(16 - x²) is the set of all non-negative real numbers.

Range: [0, ∞)

Summary:
- The domain of the function f(x) = √(16 - x²) is -4 ≤ x ≤ 4, and the range is [0, ∞).

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