The function f is defined by \(\mathrm{f(x) = 4x - 5}\). If \(\mathrm{f(k) = 23}\), what is the value of...

1. TRANSLATE the problem information

• Given information:
  • Function definition: \(\mathrm{f(x) = 4x - 5}\)
  • Function value: \(\mathrm{f(k) = 23}\)
  • Need to find: value of k
• What this tells us: We need to substitute k into the function and set it equal to 23

2. TRANSLATE into an equation

• Since \(\mathrm{f(k) = 23}\) and \(\mathrm{f(x) = 4x - 5}\), we substitute k for x:
  \(\mathrm{f(k) = 4k - 5}\)
• Setting this equal to the given output:
  \(\mathrm{4k - 5 = 23}\)

3. SIMPLIFY by solving the linear equation

• Add 5 to both sides to isolate the term with k:
  \(\mathrm{4k - 5 + 5 = 23 + 5}\)
  \(\mathrm{4k = 28}\)
• Divide both sides by 4 to solve for k:
  \(\mathrm{4k ÷ 4 = 28 ÷ 4}\)
  \(\mathrm{k = 7}\)

Answer: B) 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may try plugging answer choices back into the original function instead of setting up an equation to solve algebraically.

They might calculate \(\mathrm{f(5) = 4(5) - 5 = 15}\), \(\mathrm{f(7) = 4(7) - 5 = 23}\), etc., which works but is inefficient and can lead to computational errors or missed patterns.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic mistakes in the equation-solving process.

For example, when adding 5 to both sides, they might get \(\mathrm{4k = 18}\) instead of \(\mathrm{4k = 28}\), leading to \(\mathrm{k = 4.5}\), which isn't among the choices. This leads to confusion and guessing.

The Bottom Line:

This problem tests whether students can connect function notation with equation-solving skills. The key insight is recognizing that \(\mathrm{f(k) = 23}\) creates an equation that can be solved directly, rather than using trial-and-error with the answer choices.