Question
\:Find\:domain\:and\:range\:f\left(x\right)=x^2-2x+2
Answers
GPT 3.5
The domain of a function is the set of all possible input values (x-values) for the function.
For the given function f(x) = x^2 - 2x + 2, there are no restrictions on the domain because all real numbers can be squared, subtracted, and added.
Therefore, the domain of f(x) is all real numbers, or (-∞, ∞).
The range of a function is the set of all possible output values (y-values) for the function.
To find the range, we can consider the vertex of the parabola formed by the function. The vertex can be found using the formula x = -b/2a, where a, b, and c are the coefficients in the quadratic equation.
In this case, a = 1, b = -2, and c = 2. Plugging these values into the formula, we get x = -(-2)/(2*1) = 1.
The vertex is located at x = 1. Plugging x = 1 back into the original function, we get f(1) = 1^2 - 2(1) + 2 = 1 - 2 + 2 = 1.
So the vertex is (1, 1), which means the range of f(x) is {1}.
Therefore, the range of f(x) is {1} and the domain is all real numbers, or (-∞, ∞).
For the given function f(x) = x^2 - 2x + 2, there are no restrictions on the domain because all real numbers can be squared, subtracted, and added.
Therefore, the domain of f(x) is all real numbers, or (-∞, ∞).
The range of a function is the set of all possible output values (y-values) for the function.
To find the range, we can consider the vertex of the parabola formed by the function. The vertex can be found using the formula x = -b/2a, where a, b, and c are the coefficients in the quadratic equation.
In this case, a = 1, b = -2, and c = 2. Plugging these values into the formula, we get x = -(-2)/(2*1) = 1.
The vertex is located at x = 1. Plugging x = 1 back into the original function, we get f(1) = 1^2 - 2(1) + 2 = 1 - 2 + 2 = 1.
So the vertex is (1, 1), which means the range of f(x) is {1}.
Therefore, the range of f(x) is {1} and the domain is all real numbers, or (-∞, ∞).