Combine studied function types arithmetically to build models.
Form a function difference, domain, and sign comparison
Problem
For \(f(x)~=~x^2~+~10\) and \(g(x)~=~3x\), find \((f~-~g)(x)\), its shared domain, and the meaning of positive, zero, and negative outputs.
Big Picture
What this problem is really about
A difference of functions is an ordered comparison: evaluate both at the same input and subtract the entire second output from the first. Parentheses protect every sign when the second expression is distributed. After simplifying, the sign of the resulting function tells which original output is larger at each input, while the domain comes from their overlap.
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