This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To find the domain and range of the function f(x) = sqrt(9 - x^2): Step 1: Find the Domain The domain of a function is the set of all possible input values (x-values) for which the function is defined. For a square root function, the expression under the square root must be non-negative. So, we must have: 9 - x^2 0 Add x^2 to both sides: 9 x^2 x^2 9 Take the square root of both sides. Remember to consider both positive and negative roots: sqrt(x^2) sqrt(9) |x| 3 This inequality means that x must be between -3 and 3, inclusive. -3 x 3 In interval notation, the domain is [-3, 3]. Step 2: Find the Range The range of a function is the set of all possible output values (y-values). Let y = f(x). y = sqrt(9 - x^2) Since y is the result of a square root, y must be non-negative: y 0 Now, consider the expression 9 - x^2. The maximum value of 9 - x^2 occurs when x^2 is at its minimum. Since x^2 0, the minimum value of x^2 is 0, which happens when x=0. When x=0: y = sqrt(9 - 0^2) = sqrt(9) = 3 So, the maximum value of y is 3. The minimum value of 9 - x^2 occurs when x^2 is at its maximum. From the domain, the maximum value of x^2 is 3^2 = 9 (when x=3 or x=-3). When x=3 or x=-3: y = sqrt(9 - 3^2) = sqrt(9 - 9) = sqrt(0) = 0 So, the minimum value of y is 0. Combining these, the range of y is from 0 to 3, inclusive. 0 y 3 In interval notation, the range is [0, 3].