Let \(\mathrm{f(x) = 2x + 3}\) and \(\mathrm{g(x) = x - 4}\). Let a = 5. What is the value...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = 2x + 3}\)
  • \(\mathrm{g(x) = x - 4}\)
  • \(\mathrm{a = 5}\)
• We need to find: \(\mathrm{f(a - 2) + 2g(a)}\)

2. INFER the solution strategy

• We need to work from the inside out: first find the input values, then evaluate each function, then combine the results
• Key insight: \(\mathrm{f(a - 2)}\) means we substitute \(\mathrm{(a - 2)}\) into function f, and \(\mathrm{2g(a)}\) means we multiply \(\mathrm{g(a)}\) by 2

3. SIMPLIFY the first substitution

• Calculate the input for f: \(\mathrm{a - 2 = 5 - 2 = 3}\)
• So we need \(\mathrm{f(3)}\)

4. SIMPLIFY the function evaluations

• Evaluate \(\mathrm{f(3)}\):
  \(\mathrm{f(3) = 2(3) + 3}\)
  \(\mathrm{= 6 + 3}\)
  \(\mathrm{= 9}\)
• Evaluate \(\mathrm{g(5)}\): \(\mathrm{g(5) = 5 - 4 = 1}\)
• Calculate \(\mathrm{2g(a)}\): \(\mathrm{2g(5) = 2(1) = 2}\)

5. SIMPLIFY the final calculation

• Add the results: \(\mathrm{f(a - 2) + 2g(a) = 9 + 2 = 11}\)

Answer: B (11)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students evaluate the wrong function inputs, such as calculating \(\mathrm{f(a)}\) instead of \(\mathrm{f(a-2)}\), or calculating \(\mathrm{2g(a-2)}\) instead of \(\mathrm{2g(a)}\).

For example, if they calculate \(\mathrm{f(5) = 2(5) + 3 = 13}\) and \(\mathrm{g(3) = 3 - 4 = -1}\), then \(\mathrm{2g(3) = -2}\), giving \(\mathrm{f(a) + 2g(a-2) = 13 + (-2) = 11}\). Coincidentally, this still gives 11, so this particular error wouldn't show up. But if they calculated \(\mathrm{f(5) + g(3) = 13 + (-1) = 12}\), this may lead them to select Choice C (12).

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors during function evaluation or forget to multiply \(\mathrm{g(a)}\) by 2.

For instance, if they correctly find \(\mathrm{f(3) = 9}\) and \(\mathrm{g(5) = 1}\), but then calculate \(\mathrm{f(a-2) + g(a) = 9 + 1 = 10}\) (forgetting the factor of 2), this may lead them to select Choice A (10).

The Bottom Line:

Success requires careful attention to what's inside each function and systematic evaluation of each piece before combining results.