The function f is defined by \(\mathrm{f(x) = 25x + 30}\). What is the value of \(\mathrm{f(x)}\) when x =...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = 25x + 30}\)
  • Need to find: \(\mathrm{f(x)}\) when \(\mathrm{x = 2}\)
• What this tells us: We need to substitute \(\mathrm{x = 2}\) into the function and calculate the result

2. TRANSLATE what "\(\mathrm{f(x)}\) when \(\mathrm{x = 2}\)" means

• This means we need to find \(\mathrm{f(2)}\)
• Replace every \(\mathrm{x}\) in the function with \(\mathrm{2}\)

3. SIMPLIFY through substitution and calculation

• \(\mathrm{f(2) = 25(2) + 30}\)
• Following order of operations, multiply first: \(\mathrm{25(2) = 50}\)
• Then add: \(\mathrm{f(2) = 50 + 30 = 80}\)

Answer: C. 80

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students forget to add 30 after calculating \(\mathrm{25(2) = 50}\)

They calculate the first part correctly but stop there, thinking \(\mathrm{f(2) = 50}\). This incomplete calculation leads them to select Choice A (50).

Second Most Common Error:

Poor order of operations understanding: Students treat the expression as \(\mathrm{25 + 2 + 30}\) instead of \(\mathrm{25(2) + 30}\)

They add all numbers together: \(\mathrm{25 + 2 + 30 = 57}\), not recognizing that \(\mathrm{25x}\) means 25 times \(\mathrm{x}\). This leads them to select Choice B (57).

Third Most Common Error:

Incorrect grouping during SIMPLIFY: Students calculate \(\mathrm{(25 + 30)(2)}\) instead of \(\mathrm{25(2) + 30}\)

They add first: \(\mathrm{25 + 30 = 55}\), then multiply by 2: \(\mathrm{55 \times 2 = 110}\). This leads them to select Choice D (110).

The Bottom Line:

Function evaluation problems require careful attention to both substitution and order of operations. The key is systematically replacing the variable with the given value, then following arithmetic rules precisely.