Find the domain of the function f(x) = x^2 - 2x + 9, and give the answer in interval notation.

Step-by-step solution

1. 1

   Ask what could ever go wrong. Finding a domain is a hunt for illegal operations, and in ordinary algebra there are only three of them: dividing by zero, taking an even root of a negative number, and taking a logarithm of a non-positive number.

2. 2

   Check the rule against that list. $f(x)=x^2-2x+9$ is a polynomial: it is built from $x$ using multiplication and addition only. There is no denominator, no radical and no logarithm, so none of the three failure modes can occur.

3. 3

   Conclude for every real input. Substituting any real $x$ produces a real output — for example $f(0)=9$, $f(-100)=10209$, $f(3.5)=14.25$. Hence

   $\text{domain}(f)=(-\infty,\infty).$

4. 4

   Do not confuse this with the range. Completing the square gives

   $f(x)=(x-1)^{2}+8,$

   so the outputs never drop below $8$; equivalently the discriminant $(-2)^2-4(1)(9)=-32$ is negative, so $f$ has no real zeros. That restricts the range to $[8,\infty)$ and says nothing at all about which inputs are allowed.

5. 5

   State the general rule. Every polynomial function, of any degree, has domain $(-\infty,\infty)$. Domains only become interesting when the expression is a quotient, an even root, or a logarithm.