Find the domain of the following piecewise function. 02 2 <*24 f(x) = X+3 # 4 <X<8 ex X=8 0[2.8) 0(4.8] 3(2.8] 014.8)
The Correct Answer and Explanation is:
The correct answer is (2, 8].
Explanation
The domain of a function represents all possible input values (x-values) for which the function is defined. In a piecewise function, the domain is the union, or combination, of the individual domains specified for each piece. To find the total domain of the given function f(x), we need to analyze the conditions for each of its three parts.
- First Piece: The function is f(x) = 2 when 2 < x <= 4. This condition means the function is defined for all real numbers strictly greater than 2, up to and including 4. In interval notation, this part of the domain is written as (2, 4]. The parenthesis next to the 2 signifies that 2 is not included, while the square bracket next to the 4 signifies that 4 is included.
- Second Piece: The function is f(x) = x + 3 when 4 < x < 8. This condition means the function is defined for all real numbers strictly between 4 and 8. Neither 4 nor 8 is included in this interval. In interval notation, this domain is written as (4, 8).
- Third Piece: The function is f(x) = 2x when x = 8. This condition defines the function for a single, specific x-value, which is 8. As a set, this is simply {8}.
To determine the overall domain of f(x), we combine these three sets of x-values: (2, 4] U (4, 8) U {8}.
Let’s visualize this on a number line. The first interval (2, 4] starts just after 2 and ends at 4, including 4. The second interval (4, 8) starts just after 4 and continues up to 8, without including 8. Since the first interval includes 4, there is no gap in the domain at x=4. Combining these first two intervals gives us all numbers from 2 to 8, excluding 2 and 8, which is the interval (2, 8).
Now, we must also include the domain from the third piece, which is the single point {8}. When we take the union of the interval (2, 8) and the point {8}, we are essentially adding the endpoint 8 to our set. This action closes the interval at the upper end.
Therefore, the final domain of the entire piecewise function is all numbers greater than 2, up to and including 8. This is correctly represented in interval notation as (2, 8].
