The function f is defined by \(\mathrm{f(x) = \frac{7}{10}x + 55}\). What is the value of \(\mathrm{f(20)}\)?

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = \frac{7}{10}x + 55}\)
  • Need to find: \(\mathrm{f(20)}\)
• What this tells us: We need to substitute 20 for x in the function

2. SIMPLIFY through substitution and arithmetic

• Substitute \(\mathrm{x = 20}\) into \(\mathrm{f(x) = \frac{7}{10}x + 55}\):
  \(\mathrm{f(20) = \frac{7}{10}(20) + 55}\)

• Calculate \(\mathrm{\frac{7}{10} × 20}\):
  \(\mathrm{\frac{7}{10} × 20 = \frac{7 × 20}{10} = \frac{140}{10} = 14}\)

• Add the constant term:
  \(\mathrm{f(20) = 14 + 55 = 69}\)

Answer: 69

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual gap with function notation: Students may not understand what \(\mathrm{f(20)}\) means and might try to solve for \(\mathrm{x = 20}\) instead of substituting 20 into the function.

This confusion about function evaluation vs equation solving leads to getting stuck and guessing.

Second Most Common Error:

Weak SIMPLIFY execution: Students make arithmetic errors when multiplying the fraction \(\mathrm{\frac{7}{10}}\) by 20, perhaps calculating it as \(\mathrm{7 × 2 = 14}\) instead of \(\mathrm{\frac{7 × 20}{10} = 14}\), or getting confused with fraction operations.

This leads to incorrect numerical answers like 69 ± calculation errors.

The Bottom Line:

This problem tests whether students truly understand function notation - that \(\mathrm{f(20)}\) means "substitute 20 for x" rather than "solve when the function equals 20." The arithmetic is straightforward once they know what operation to perform.