What are the domain and range of g(x)=−14(x−17)2+61𝑔(𝑥)=-14(𝑥-17)2+61? A. Domain: All real numbersRange: g(x)≤61𝑔(𝑥)≤61 B. Domain: x≤17𝑥≤17Range: g(x)≤61𝑔(𝑥)≤61 C. Domain: All real numbersRange: x≤17𝑥≤17 D. Domain: g(x)≥61𝑔(𝑥)≥61Range: x≤17
Question
What are the domain and range of g(x)=−14(x−17)2+61𝑔(𝑥)=-14(𝑥-17)2+61? A. Domain: All real numbersRange: g(x)≤61𝑔(𝑥)≤61 B. Domain: x≤17𝑥≤17Range: g(x)≤61𝑔(𝑥)≤61 C. Domain: All real numbersRange: x≤17𝑥≤17 D. Domain: g(x)≥61𝑔(𝑥)≥61Range: x≤17
Solution
The correct answer is A. Domain: All real numbers Range: g(x)≤61.
Here's why:
The domain of a function is the set of all possible input values (often 'x' values), which will produce a valid output from a particular function. The given function g(x) = -14(x - 17)^2 + 61 is a quadratic function, and the domain of a quadratic function is all real numbers. So, the domain of this function is all real numbers.
The range of a function is the set of possible output values (often 'y' values), which are valid for a given input to the function. In this case, the function g(x) = -14(x - 17)^2 + 61 is a downward-opening parabola (because the coefficient of x^2 is negative), and the vertex of the parabola is (17, 61). This means that the maximum value of the function is 61, and it will decrease as x moves away from 17 in either direction. So, the range of this function is g(x) ≤ 61.
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