Question

What are the domain and range of f(x) = ½ (x+2)² + 1 ?

a. domain: x > 1; range: y > 2
b. domain: x > 1; range: all real numbers
c. domain: all real numbers; range: y > 1
d. domain: all real numbers; range: all real numbers

Answer 1

The domain of the function f(x) = ½(x+2)² + 1 is all real numbers, and the range is y > 1, because the parabola opens upwards and the vertex represents the minimum y-value of the function.

Step-by-step explanation:

The domain of a function is the set of all possible input values (x-values) for which the function is defined, and the range is the set of all possible output values (y-values) that the function can take. For the function f(x) = ½ (x+2)² + 1, the domain is all real numbers because a quadratic function is defined for all x-values. The range, however, is restricted by the vertex of the parabola. Since the quadratic term is positive (½ > 0), the parabola opens upwards and the vertex represents the minimum value of the function. The vertex in this case is at x = -2 which gives a minimum y-value of 1 when plugged into the function. Therefore, the range of the function is y > 1.