Determine the domain and range of the quadratic function. (Enter your answers using interval notation.) $$ f(x)=(x-3)^{2}+6 $$ domain range

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**Step 1: Domain**

The function is 
$$
f(x)=(x-3)^2+6.
$$
Since this is a quadratic function (with a squared term) and there are no restrictions such as division by zero or a square root of a negative number, the function is defined for all real values of $$ x $$.

Thus, the **domain** is:
$$
(-\infty, \infty)
$$

**Step 2: Range**

A quadratic function in the form 
$$
f(x)=(x-h)^2+k
$$
has its vertex at $$(h,k)$$. The given function has $$ h = 3 $$ and $$ k = 6 $$. Since the coefficient of the squared term is positive (implicitly $$ 1 $$), the parabola opens upward.

The vertex $$(3,6)$$ represents the minimum point of the function. Therefore, the smallest value $$ f(x) $$ can take is $$ 6 $$, and as $$ x $$ moves away from $$ 3 $$, $$ f(x) $$ increases without bound.

Thus, the **range** is:
$$
[6, \infty)
$$
**Domain:** $$(-\infty, \infty)$$ **Range:** $$[6, \infty)$$

Determine the domain and range of the quadratic function. (Enter your answers using interval notation.) \[ f(x)=(x-3)^{2}+6 \] domain range

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