Given f(x) = 5x + 2 and g(x) = x - 6, find (f + g)(x) and state its domain.

Step-by-step solution

1. 1

   Write down what the plus sign between two functions means. Adding functions is defined pointwise: for every input $x$ you evaluate both rules and add the outputs. In symbols,

   $(f+g)(x)=f(x)+g(x).$

   There is no new operation to learn here — the only work is substituting the two formulas.

2. 2

   Substitute both rules, keeping each one in brackets. The brackets matter as soon as a subtraction appears later, so build the habit now:

   $(f+g)(x)=(5x+2)+(x-6).$

3. 3

   Drop the brackets and group like terms. Because every bracket is preceded by a plus sign, nothing changes sign:

   $(5x+2)+(x-6)=\underbrace{5x+x}_{\text{x-terms}}+\underbrace{2-6}_{\text{constants}}.$

4. 4

   Combine each group. The coefficient of $x$ is $5+1=6$, and the constants give $2-6=-4$:

   $(f+g)(x)=6x-4.$

5. 5

   Find the domain from the domains of the pieces, not from the answer. A sum is defined exactly where both originals are defined. Both $f$ and $g$ are polynomials, defined for every real number, so

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

   This order matters: for quotients such as $f/g$ the simplified formula can hide a value that must still be excluded, so always read the domain off the original pieces.