What is the domain and range of the relation y = 2(x-5)^2 + 8? Express in set notation.

---------------------------------------
Domain: x E R
Range : y E R
---------------------------------------
Thanks!

this is a parabola in standard position with vertex

(5,8) opening upwards

so domain: reals
range: reals, y ≥ 8

To determine the domain and range of the relation y = 2(x-5)^2 + 8, we can first look at the characteristics of the equation.

The quadratic term, (x-5)^2, indicates that the value inside the parentheses can be any real number. This means that x can take on any value from negative infinity to positive infinity, so the domain of the relation is D = (-∞,∞).

Next, we consider the range. Since the equation involves squaring the term inside the parentheses, the result will always be greater than or equal to 0. Multiplying by 2 and adding 8 further increases the value. Hence, the range will be all real numbers greater than or equal to 8. Therefore, the range of the relation is R = [8, ∞).

In set notation, we represent the domain and range as follows:
Domain: D = (-∞, ∞)
Range: R = [8, ∞)

Remember that parentheses "(" and ")" indicate open intervals (values are not included), brackets "[" and "]" indicate closed intervals (values are included), and ∞ represents infinity.