The function g is defined as \(\mathrm{g(x) = 5x + a}\), where a is a constant. If \(\mathrm{g(4) = 31}\),...

1. TRANSLATE the given information

• Given information:
  • Function: \(\mathrm{g(x) = 5x + a}\) (where a is unknown)
  • Condition: \(\mathrm{g(4) = 31}\)
• What this tells us: We need to use the condition to find the value of a

2. TRANSLATE the condition into an equation

• Since \(\mathrm{g(4) = 31}\), substitute \(\mathrm{x = 4}\) into our function:
  \(\mathrm{g(4) = 5(4) + a}\)
• This gives us: \(\mathrm{31 = 5(4) + a}\)

3. SIMPLIFY to solve for a

• Calculate: \(\mathrm{5(4) = 20}\)
• So our equation becomes: \(\mathrm{31 = 20 + a}\)
• Isolate a: \(\mathrm{a = 31 - 20 = 11}\)

Answer: C. 11

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may substitute incorrectly or set up the wrong equation. For example, they might write \(\mathrm{g(4) = 5 + 4a}\) instead of \(\mathrm{g(4) = 5(4) + a}\), misunderstanding how function evaluation works.

This leads to the wrong equation \(\mathrm{31 = 5 + 4a}\), giving \(\mathrm{a = 26/4 = 6.5}\), which doesn't match any answer choice and causes confusion and guessing.

Second Most Common Error:

Arithmetic errors during SIMPLIFY: Students correctly set up \(\mathrm{31 = 20 + a}\) but make calculation mistakes. They might compute \(\mathrm{5(4) = 24}\) instead of 20, leading to \(\mathrm{31 = 24 + a}\), so \(\mathrm{a = 7}\). Or they might make an error in the final subtraction.

Since 7 isn't among the choices, this leads to confusion, or they might select the closest value and pick Choice C (11) by coincidence.

The Bottom Line:

This problem tests whether students can properly translate function notation into algebraic equations. The math itself is straightforward once the setup is correct - the challenge lies in correctly interpreting what \(\mathrm{g(4) = 31}\) means in terms of the given function definition.