Graph the following piecewise function and then find the domain
The Correct Answer and Explanation is:
The correct answer is (-4,9).
Explanation:
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For a piecewise function, the domain is the union of the domains of all its individual pieces. The given function is:
f(x) = { 3x² + 1 if -4 < x < 6
{ 6 if 6 ≤ x < 9
We need to analyze the conditions for each piece to determine the total set of x-values.
1. Analyze the First Piece:
The first rule, f(x) = 3x² + 1, is defined for the interval specified by the inequality -4 < x < 6. This means the function follows this quadratic rule for all x-values that are strictly greater than -4 and strictly less than 6. The endpoints, -4 and 6, are not included in this portion of the domain. In interval notation, this domain is represented as (-4, 6). The parentheses signify that the endpoints are excluded.
2. Analyze the Second Piece:
The second rule, f(x) = 6, is a constant function defined for the interval 6 ≤ x < 9. This means the function has a value of 6 for all x-values that are greater than or equal to 6 but strictly less than 9. In this case, the endpoint 6 is included, which is indicated by the “less than or equal to” sign (≤). The endpoint 9 is excluded. In interval notation, this domain is represented as [6, 9). The square bracket [ indicates that 6 is included, while the parenthesis ) indicates that 9 is excluded.
3. Combine the Domains:
To find the total domain of the entire piecewise function, we must combine, or take the union of, the two individual domains:
Domain = (-4, 6) U [6, 9)
The first interval, (-4, 6), covers all numbers from -4 up to 6, but not including 6. The second interval, [6, 9), starts exactly at 6 (including it) and goes up to 9 (not including it). When we combine these two sets, the gap at x = 6 from the first interval is perfectly filled by the second interval. This creates a single, continuous interval. The combined domain starts where the first interval begins and ends where the second interval stops.
Thus, the overall domain starts at -4 (not included) and extends to 9 (not included). In interval notation, the final domain is (-4, 9).
