Find the domain of the given function. Write your answers in both set builder and interval notation, also with absolute value. : f(x)= `sqrt(x^2 + 12x + 36) `

The domain of a function is a set of all values of x for which the function is defined.

For a function involving square roots, as the given function, the domain might be restricted by the fact that the expression under the square root has to be nonnegative (positive or zero), because the square root is not defined for negative numbers.

Consider the expression under the square root in the given function:

`x^2 + 12x + 36`

This can be factored as `x^2 + 12x + 36 = (x+6)^2` . Since this is a square of a real number, it is always nonnegative. Therefore, any value of x will produce a positive (or zero) value of the expression under the square root, so the function is defined for any real x.

This function can also be rewritten as

`f(x) = sqrt((x+6)^2)=|x+6|.`

The domain of this function is all real numbers.

In set-builder notation, it is written as `{x | x in R}`

In interval notation, it is written as `(-oo, oo).```

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