\(\mathrm{f(x) = \frac{x + 7}{4}}\) For the function f defined above, what is the value of \(\mathrm{f(9) - f(1)}\)?...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = \frac{x + 7}{4}}\)
  • Need to find: \(\mathrm{f(9) - f(1)}\)
• This means we need to substitute \(\mathrm{x = 9}\) and \(\mathrm{x = 1}\) separately into the function, then subtract the results

2. SIMPLIFY to find f(9)

• Substitute \(\mathrm{x = 9}\) into \(\mathrm{f(x) = \frac{x + 7}{4}}\):

\(\mathrm{f(9) = \frac{9 + 7}{4}}\)
\(\mathrm{= \frac{16}{4}}\)
\(\mathrm{= 4}\)

3. SIMPLIFY to find f(1)

• Substitute \(\mathrm{x = 1}\) into \(\mathrm{f(x) = \frac{x + 7}{4}}\):

\(\mathrm{f(1) = \frac{1 + 7}{4}}\)
\(\mathrm{= \frac{8}{4}}\)
\(\mathrm{= 2}\)

4. SIMPLIFY the final calculation

• Calculate \(\mathrm{f(9) - f(1)}\):

\(\mathrm{f(9) - f(1) = 4 - 2 = 2}\)

Answer: B. 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Making arithmetic errors during substitution or calculation

Students might incorrectly calculate \(\mathrm{\frac{9 + 7}{4}}\) as something other than 4, or \(\mathrm{\frac{1 + 7}{4}}\) as something other than 2. For example, if they calculate \(\mathrm{f(9) = \frac{9}{4}}\) (forgetting to add 7) and \(\mathrm{f(1) = 0}\) (forgetting to add 7), they would get \(\mathrm{f(9) - f(1) = \frac{9}{4} - 0 = \frac{9}{4}}\).

This may lead them to select Choice D \(\mathrm{(\frac{9}{4})}\)

Second Most Common Error:

Poor TRANSLATE reasoning: Misunderstanding what the function notation means

Some students might try to substitute \(\mathrm{9 - 1 = 8}\) into the function instead of finding \(\mathrm{f(9)}\) and \(\mathrm{f(1)}\) separately, or they might get confused about the order of operations within the function.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem tests whether students can systematically work with function notation and perform accurate arithmetic. Success requires careful substitution and methodical calculation rather than rushing through the steps.