The function f is defined by \(\mathrm{f(x) = x^2 + 2x - 5}\). What is the value of \(\mathrm{f(-3)}\)?

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = x^2 + 2x - 5}\)
  • Need to find: \(\mathrm{f(-3)}\)
• What this tells us: We need to substitute \(\mathrm{x = -3}\) into the function expression

2. SIMPLIFY through substitution and calculation

• Substitute -3 for every x in the expression:

\(\mathrm{f(-3) = (-3)^2 + 2(-3) - 5}\)

• Apply order of operations (exponents first):

\(\mathrm{f(-3) = 9 + 2(-3) - 5}\)

• Continue with multiplication:

\(\mathrm{f(-3) = 9 + (-6) - 5}\)

• Combine terms from left to right:

\(\mathrm{f(-3) = 9 - 6 - 5 = 3 - 5 = -2}\)

Answer: -2

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about exponent rules: Students incorrectly think that \(\mathrm{(-3)^2 = -9}\) instead of 9.

This happens because they confuse \(\mathrm{(-3)^2}\) with \(\mathrm{-(3)^2}\), not realizing that when the negative sign is inside the parentheses, it gets squared along with the 3. With this error: \(\mathrm{f(-3) = -9 + (-6) - 5 = -20}\).

This may lead them to select Choice A (-20).

Second Most Common Error:

Weak SIMPLIFY execution: Students make sign errors when combining the terms, particularly when dealing with multiple negative numbers.

For example, they might calculate \(\mathrm{9 + (-6) - 5}\) as \(\mathrm{9 + 6 - 5 = 10}\), or make other arithmetic mistakes with the negative signs. Various sign errors can lead to different incorrect results.

This may lead them to select Choice B (-14) or Choice D (2).

The Bottom Line:

Function evaluation problems require careful attention to both the substitution process and arithmetic with negative numbers. The key is treating \(\mathrm{(-3)^2}\) as "negative three, quantity squared" which equals positive 9.