The function f is defined by \(\mathrm{f(x) = 5x + 8}\). For what value of x does \(\mathrm{f(x) = 58}\)?

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = 5x + 8}\)
  • We need: the value of x when \(\mathrm{f(x) = 58}\)
• What this tells us: We need to solve an equation where the function output equals 58

2. TRANSLATE to set up the equation

• Since \(\mathrm{f(x) = 5x + 8}\) and we want \(\mathrm{f(x) = 58}\), we substitute:
  \(\mathrm{58 = 5x + 8}\)
• This gives us a linear equation to solve

3. SIMPLIFY through algebraic steps

• Start with: \(\mathrm{58 = 5x + 8}\)
• Subtract 8 from both sides: \(\mathrm{58 - 8 = 5x + 8 - 8}\)
• This gives us: \(\mathrm{50 = 5x}\)
• Divide both sides by 5: \(\mathrm{50 \div 5 = 5x \div 5}\)
• Final result: \(\mathrm{10 = x}\)

4. Check your answer

• Substitute \(\mathrm{x = 10}\) back into \(\mathrm{f(x) = 5x + 8}\)
• \(\mathrm{f(10) = 5(10) + 8}\)
  \(\mathrm{= 50 + 8}\)
  \(\mathrm{= 58}\) ✓

Answer: A. 10

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misinterpret what the question is asking and calculate \(\mathrm{f(58)}\) instead of solving \(\mathrm{f(x) = 58}\).

They think: "The function is \(\mathrm{f(x) = 5x + 8}\), and I see 58, so \(\mathrm{f(58) = 5(58) + 8}\)
\(\mathrm{= 290 + 8}\)
\(\mathrm{= 298}\)."

This may lead them to select Choice D (298).

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic mistakes during the solving process, particularly when subtracting 8 from 58.

They might calculate: \(\mathrm{58 - 8 = 52}\) instead of 50, leading to \(\mathrm{x = 52 \div 5 = 10.4}\), then guess among the remaining choices.

This causes them to get stuck and guess, potentially selecting Choice B (13) or Choice C (50).

The Bottom Line:

This problem tests whether students can translate function language into algebraic equations and then solve systematically. The key insight is recognizing that "\(\mathrm{f(x) = 58}\)" means setting the entire function expression equal to 58, not plugging 58 into the function.