What is the domain of the function
f(x)=11/Squareroot of 4-x^2
a.(-11,11)
b.[-11,11]
c. (-2,2)
d. [-2,2]
What is the range of the function?
y=8/Squareroot of 4-x^2
a. [4, infinity)
b. (-infinity, 4]
c. [8, infinity)
d. (-infinity, 12]
e. [6, infinity)
f. (-infinity, 6]
To determine the domain and range of a function, we need to consider any restrictions on the input variable (x) and the resulting output values (y).
Let's start with the domain of the function f(x) = 11/√(4-x^2).
The function involves division by the square root of (4-x^2). Division by zero is undefined, so we need to find out if there are any values of x that would make the denominator equal to zero.
To do that, we set the square root of (4-x^2) equal to zero and solve for x:
√(4-x^2) = 0
Squaring both sides of the equation, we get:
4 - x^2 = 0
Rearranging the equation:
x^2 = 4
Taking the square root of both sides:
x = ±2
So, the values of x that would make the denominator zero are x = -2 and x = 2.
However, since the denominator is the square root of (4-x^2), the function is undefined for x values that make the expression under the square root negative. In this case, if 4 - x^2 < 0, since the square root of a negative number is not a real number.
To determine when 4 - x^2 < 0, we can rewrite it as x^2 > 4. Taking the square root of both sides, we get |x| > 2. This means that the function is only defined when |x| ≤ 2.
Therefore, the domain of the function is the interval [-2, 2] or option (d) in the given choices.
Moving on to the range of the function y = 8/√(4-x^2):
To find the range, we need to consider the possible values of y.
Since the denominator is the square root of (4-x^2), the function is defined as long as the expression under the square root is not negative. This occurs when 4 - x^2 ≥ 0.
Solving for x^2:
x^2 ≤ 4
Taking the square root of both sides:
|x| ≤ 2
Now, let's consider the numerator, which is a constant (8). As long as the denominator is not zero, changing the value of x will not change the numerator.
Therefore, the range of the function y = 8/√(4-x^2) is determined by the denominator, which is the square root of (4-x^2):
Since the denominator is always positive or zero, the function's range is all values greater than or equal to 0.
So, the range is [0, ∞) or option (a) in the given choices.