What is the domain of the function f(x)=11/Square root of 4-x^2
What is the range of the function
y=8/Square root of 4-x^2
Show your steps too.
domain is all reals except where the denominator is zero
so, all reals except 2 and -2
range is all reals except 0<= y < 11/2
To determine the domain and range of the given function, we need to consider the values of x that make the function well-defined.
For the function f(x) = 11/√(4 - x^2):
Step 1: The function involves a square root, so the expression inside the square root (4 - x^2) should be non-negative (≥ 0). Set up the inequality:
4 - x^2 ≥ 0
Step 2: Solve the inequality:
x^2 ≤ 4
Step 3: Take the square root of both sides:
√(x^2) ≤ √4
Step 4: Simplify:
|x| ≤ 2
Step 5: This implies that -2 ≤ x ≤ 2. Therefore, the domain of the function f(x) is the closed interval [-2, 2].
For the function y = 8/√(4 - x^2):
Step 1: Similar to the previous case, the expression inside the square root (4 - x^2) should be non-negative (≥ 0). Set up the inequality:
4 - x^2 ≥ 0
Step 2: Solve the inequality:
x^2 ≤ 4
Step 3: Take the square root of both sides:
√(x^2) ≤ √4
Step 4: Simplify:
|x| ≤ 2
Step 5: The range of the function is determined by the numerator, which is a constant (8), and the denominator, which can vary based on x. The denominator, √(4 - x^2), can take any positive value. Therefore, the range of the function y is (0, ∞), which represents all non-negative real numbers.
To summarize:
- For the function f(x) = 11/√(4 - x^2), the domain is [-2, 2].
- For the function y = 8/√(4 - x^2), the range is (0, ∞).