If f(x) = 2 - x^2 and g(x) = x^2 + 4x - 60, what is the domain of f - g? Give the answer in interval notation.

Step-by-step solution

1. 1

   Recall the rule for the domain of a combination. For any of $f+g$, $f-g$ or $fg$, the domain is the intersection of the two individual domains:

   $\operatorname{dom}(f-g)=\operatorname{dom}(f)\cap\operatorname{dom}(g)$

   (The quotient $f/g$ is the exception — it additionally removes the zeros of $g$.)

2. 2

   Find each domain separately. $f(x)=2-x^2$ is a polynomial: it involves only multiplication and addition of real numbers, with no division by a variable and no even root. So

   $\operatorname{dom}(f)=(-\infty,\infty)$

   The same reasoning applies to $g(x)=x^2+4x-60$:

   $\operatorname{dom}(g)=(-\infty,\infty)$

3. 3

   Intersect the two domains.

   $(-\infty,\infty)\cap(-\infty,\infty)=(-\infty,\infty)$

4. 4

   Confirm by forming the difference explicitly. Subtract, distributing the minus sign across every term of $g$:

   $(f-g)(x)=\left(2-x^2\right)-\left(x^2+4x-60\right)=-2x^2-4x+62$

   That result is again a polynomial, so there is nothing anywhere on the real line that could make it undefined.

5. 5

   State the answer. The domain of $f-g$ is all real numbers:

   $(-\infty,\infty)$

   The zeros of $g$ — the solutions of $x^2+4x-60=0$, namely $x=6$ and $x=-10$ — are irrelevant here. They would only matter for $f/g$.