Answer:
[9, 9+√18.25] ≈ [9, 13.272]
Explanation:
You want the range of f(x)= 9+√(6+7x-x^2), determined algebraically.
Range
The range is the set of values that f(x) may take on. It is the vertical extent of the graph. Here, it will be limited by the possible values the radical may have.
The expression under the radical is a quadratic that opens downward. It will have a maximum at its vertex, and a minimum usable value of 0.
The vertex of the quadratic can be found by putting it in vertex form:
-x² +7x +6 = -(x² -7x) +6 = -(x² -7x +(7/2)²) +6 +(7/2)² = -(x -7/2)² +18.25
The maximum value the quadratic may have is 18.25.
Then the range of values that f(x) may have is ...
9 +0 ≤ f(x) ≤ 9 +√18.25
The range is [9, 9+√18.25] ≈ [9, 13.272].