For f(x) = 2 - x^2 and g(x) = x^2 + 4x - 60, find (f + g)(x) and determine its domain.

Step-by-step solution

1. 1

   Apply the pointwise definition of a sum of functions. For every input $x$,

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

   Both brackets are added, so no signs flip when they are removed.

2. 2

   Sort the terms by degree.

   $\underbrace{\left(-x^{2}+x^{2}\right)}_{\text{quadratic}}+\underbrace{4x}_{\text{linear}}+\underbrace{\left(2-60\right)}_{\text{constant}}.$

3. 3

   Combine each group and notice the cancellation. The quadratic terms are exact opposites:

   $-x^{2}+x^{2}=0,\qquad 2-60=-58,$

   so

   $(f+g)(x)=4x-58.$

   The sum of two quadratics is usually quadratic; here the leading coefficients happen to be $-1$ and $+1$, which is why the degree drops.

4. 4

   Read the domain from the original functions, not from the simplified answer. A sum is defined wherever both pieces are defined. Both $f$ and $g$ are polynomials, defined on all of $\mathbb{R}$, so

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

5. 5

   Check with a value. At $x=10$: $f(10)=2-100=-98$ and $g(10)=100+40-60=80$, so $f(10)+g(10)=-18$; and $4(10)-58=-18$. The formulas agree.