Question

What is the domain and range of `y=1/2x^2+4`?

Answer 1

Consider the function `y= f(x)`

The domain of this function is all the values of x for which the function holds. The range is all those values of y for which the function is valid.

Now, coming to your question.
`y = x^2 / 2 + 4`
This function is valid for any real value of x. Thus the domain of this function is the set of all real numbers, i.e. , `R`.

Now, separate out x.
`y= x^2 /2 +4`

=> `y-4 = x^2 /2`

=> `2(y-4) = x^2`

=> `{2(y-4)}^(1/2) = x`

Thus, the function is valid for all real numbers greater than or equal to 4. Therefore the range of this function is [4, `oo`).