Find domain and range of this real function f(x) = root under 9 - x^2 ...
F(x) = √(9 - x²)
• Domain:
Since there is a square root x must be defined such that,
9 - x² ≥ 0 => 3² - x² ≥ 0
(3 - x)(3 + x) ≥ 0
(x - 3)(x + 3) ≤ 0
x € [-3,3]
• Range :
The range of f(x) can be any real number
Find domain and range of this real function f(x) = root under 9 - x^2 ...
Domain and Range of f(x) = √(9-x^2)
Domain:
The domain of a function is the set of all possible values of x for which the function is defined. In this case, we need to find the values of x for which the square root of 9-x^2 is a real number.
To ensure that the square root is real, we must have 9-x^2 ≥ 0. Solving this inequality, we get:
9 - x^2 ≥ 0
x^2 ≤ 9
-3 ≤ x ≤ 3
Therefore, the domain of f(x) is [-3, 3].
Range:
The range of a function is the set of all possible values of y that the function can take. In this case, the function f(x) = √(9-x^2) can only take non-negative values, because the square root of a negative number is not a real number.
Furthermore, the maximum value of f(x) occurs when x = 0, which gives f(0) = √9 = 3. Therefore, the range of f(x) is [0, 3].
Summary:
- The domain of f(x) = √(9-x^2) is [-3, 3].
- The range of f(x) is [0, 3].
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