Function h is defined for all real numbers and satisfies \(\mathrm{h(5 - x) = 2x + 7}\) for all real...

1. TRANSLATE the problem information

• Given information:
  • Function h satisfies \(\mathrm{h(5 - x) = 2x + 7}\) for all real numbers x
  • Need to find \(\mathrm{h(1)}\)
• What this tells us: We have a functional equation where the input is \(\mathrm{(5-x)}\) and the output is \(\mathrm{2x + 7}\)

2. INFER the approach

• Key insight: To find \(\mathrm{h(1)}\), we need the input to function h to be 1
• This means we need to find what value of x makes \(\mathrm{(5-x) = 1}\)
• Once we know that x-value, we can use the given formula \(\mathrm{2x + 7}\)

3. SIMPLIFY to find the required x-value

• Set up the equation: \(\mathrm{5 - x = 1}\)
• Solve step by step:
  • \(\mathrm{5 - x = 1}\)
  • \(\mathrm{-x = 1 - 5}\)
  • \(\mathrm{-x = -4}\)
  • \(\mathrm{x = 4}\)

4. SIMPLIFY to calculate h(1)

• Substitute \(\mathrm{x = 4}\) into the formula:
  \(\mathrm{h(1) = 2(4) + 7 = 8 + 7 = 15}\)

Answer: D. 15

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the connection between finding \(\mathrm{h(1)}\) and determining what x-value makes the input equal 1. Instead, they might try to directly substitute something like \(\mathrm{x = 1}\) into the equation, getting \(\mathrm{h(5-1) = h(4) = 2(1) + 7 = 9}\), which gives them the wrong input to the function.

This may lead them to select Choice A (9)

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{5 - x = 1}\) but make algebraic errors when solving. Common mistakes include getting \(\mathrm{x = -4}\) instead of \(\mathrm{x = 4}\), or making arithmetic errors when calculating \(\mathrm{2x + 7}\).

This leads to confusion and incorrect answer selection

The Bottom Line:

The key challenge is recognizing that functional equations require you to work backwards from the desired input to find the appropriate parameter value. Students must understand that \(\mathrm{h(1)}\) means "what does h output when its input is 1?" rather than "substitute 1 for x in the given expression."