Let \(\mathrm{f(x) = \frac{5}{6}x - 9}\) and \(\mathrm{g(x) = \frac{1}{3}x + 7}\). Which of the following expressions is equivalent to...

1. TRANSLATE the problem information

• Given functions:
  • \(\mathrm{f(x) = \frac{5}{6}x - 9}\)
  • \(\mathrm{g(x) = \frac{1}{3}x + 7}\)
• Need to find: \(\mathrm{f(x) - g(x)}\)

2. TRANSLATE this into a computational expression

• Substitute the function definitions:
  \(\mathrm{f(x) - g(x) = \frac{5}{6}x - 9 - [(\frac{1}{3}x + 7)]}\)

3. SIMPLIFY by distributing the negative sign

• Apply the distributive property to remove brackets:
  \(\mathrm{= \frac{5}{6}x - 9 - \frac{1}{3}x - 7}\)
• The key is that the negative sign applies to both terms inside the brackets

4. SIMPLIFY by grouping and combining like terms

• Group x-terms and constants separately:
  \(\mathrm{= \frac{5}{6}x - \frac{1}{3}x - 9 - 7}\)

• For x-terms, find common denominator:
  \(\mathrm{\frac{5}{6} - \frac{1}{3} = \frac{5}{6} - \frac{2}{6} = \frac{3}{6} = \frac{1}{2}}\)

• For constants:
  \(\mathrm{-9 - 7 = -16}\)

5. Write the final simplified form

• \(\mathrm{f(x) - g(x) = \frac{1}{2}x - 16}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Making sign errors when distributing the negative through the parentheses

Students often write:
\(\mathrm{\frac{5}{6}x - 9 - \frac{1}{3}x + 7}\)
instead of
\(\mathrm{\frac{5}{6}x - 9 - \frac{1}{3}x - 7}\)

This gives them
\(\mathrm{-9 + 7 = -2}\)
for the constant term instead of
\(\mathrm{-9 - 7 = -16}\).

This may lead them to select Choice D (\(\mathrm{\frac{1}{2}x - 2}\))

The Bottom Line:

Function subtraction requires careful attention to sign management—the negative sign must be distributed to every term in the second function, not just the first term.