Determine the domain and range of the function.
4-X^2 if x<0
f(x)= 4-2X if x≤x≤2
sqrt(x-2) If x>2
To determine the domain and range of the function, let's analyze each piece of the function separately.
First, let's consider the piece of the function when x < 0. In this case, the function is given by f(x) = 4 - x^2. Since there are no restrictions on x, the domain for this piece of the function is all real numbers. However, since x^2 is always nonnegative, the range will be all real numbers less than or equal to 4.
Next, let's analyze the piece of the function when 0 ≤ x ≤ 2. In this case, the function is given by f(x) = 4 - 2x. Again, there are no restrictions on x, so the domain for this piece of the function is all real numbers. The range will be all real numbers less than or equal to 4 as well.
Finally, let's look at the piece of the function when x > 2. In this case, the function is given by f(x) = sqrt(x - 2). Since taking the square root is only defined for nonnegative numbers, the domain for this piece will be x greater than or equal to 2. The range will be all real numbers greater than or equal to 0.
Combining all the pieces, we can conclude that the domain of the function is all real numbers, and the range is all real numbers less than or equal to 4 for x < 0, and all real numbers greater than or equal to 0 for x ≥ 0.