For f(x) = 5x + 2 and g(x) = x − 6, find (f − g)(x) and simplify.

Step-by-step solution

1. 1

   Write down the definition of the difference of two functions. By definition

   $(f-g)(x)=f(x)-g(x)$

   The subtraction applies to the entire output of $g$, which is why the next step needs brackets.

2. 2

   Substitute, keeping g in brackets.

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

   Omitting those brackets is the single most common error, and it produces $5x+2-x-6$, which is wrong.

3. 3

   Distribute the minus sign. It changes the sign of every term inside:

   $5x+2-x+6$

4. 4

   Combine like terms. The $x$ terms give $5x-x=4x$; the constants give $2+6=8$:

   $(f-g)(x)=4x+8$

   $\boxed{4x+8}$

5. 5

   Check with a test value. At $x=3$: $f(3)=17$ and $g(3)=-3$, so $f(3)-g(3)=17-(-3)=20$; and $4(3)+8=20$. The values match, confirming the sign handling. Note also that $4x+8=4(x+2)$, so the difference has a single zero at $x=-2$.