The function f is defined by \(\mathrm{f(x)} = \frac{2\mathrm{x} + 3}{4\mathrm{x} + 5}\). What is the value of \(\mathrm{f(3)}\)? 9/17...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = \frac{2x + 3}{4x + 5}}\)
  • Need to find: \(\mathrm{f(3)}\)

• What this tells us: We need to substitute \(\mathrm{x = 3}\) everywhere x appears in the function expression

2. SIMPLIFY by substituting and calculating

• Substitute \(\mathrm{x = 3}\) into the function:

\(\mathrm{f(3) = \frac{2(3) + 3}{4(3) + 5}}\)

• Calculate the numerator: \(\mathrm{2(3) + 3 = 6 + 3 = 9}\)
• Calculate the denominator: \(\mathrm{4(3) + 5 = 12 + 5 = 17}\)

• Final result: \(\mathrm{f(3) = \frac{9}{17}}\)

3. Verify the fraction is in simplest form

• Since 17 is prime and doesn't divide 9, the fraction \(\mathrm{\frac{9}{17}}\) cannot be simplified further

Answer: A) \(\mathrm{\frac{9}{17}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when calculating the numerator or denominator

For example, they might calculate:

• \(\mathrm{2(3) + 3 = 6 + 3 = 8}\) (instead of 9), or
• \(\mathrm{4(3) + 5 = 12 + 5 = 16}\) (instead of 17)

This leads to incorrect fractions like \(\mathrm{\frac{8}{17}}\), \(\mathrm{\frac{9}{16}}\), or \(\mathrm{\frac{8}{16}}\), causing them to select wrong answer choices or feel confused when their result doesn't match any option.

The Bottom Line:

This problem tests careful arithmetic execution more than complex mathematical reasoning. The key is methodical substitution followed by accurate basic calculations - seemingly simple steps where small errors have big consequences.