Given f(x) = 2 - x^2 and g(x) = x^2 + 4x - 60, find (f - g)(x) and simplify.

Step-by-step solution

1. 1

   Translate the notation into an ordinary subtraction. By definition $(f-g)(x)=f(x)-g(x)$, so the composite symbol just means 'subtract the second formula from the first'. Write it with the whole of $g$ inside parentheses — that is the single habit that prevents the most common error here:

   $(f-g)(x)=(2-x^2)-(x^2+4x-60)$

2. 2

   Distribute the minus sign across every term of $g$. Subtraction applies to all three terms, not just the first:

   $2-x^2-x^2-4x+60$

   The sign of $-60$ becomes $+60$; writing $-60$ here is the mistake that turns the answer into $-2x^2-4x-58$.

3. 3

   Group like terms by degree. Quadratic terms: $-x^2-x^2=-2x^2$. Linear terms: only $-4x$. Constants: $2+60=62$.

4. 4

   Write the simplified difference.

   $(f-g)(x)=-2x^2-4x+62$

   Note it is still a quadratic: the $x^2$ terms did not cancel because they carry the same sign after distribution, not opposite signs.

5. 5

   Check at three values of $x$. At $x=0$: $f=2$, $g=-60$, so $f-g=62$, and the formula gives $62$ ✓. At $x=1$: $f=1$, $g=-55$, difference $56$; formula gives $-2-4+62=56$ ✓. At $x=-3$: $f=-7$, $g=9-12-60=-63$, difference $56$; formula gives $-18+12+62=56$ ✓.