For the function f defined by \(\mathrm{f(x) = \frac{a}{x} + b}\), where a and b are constants and x neq...

1. TRANSLATE the given conditions into equations

• Given information:
  • \(\mathrm{f(x) = \frac{a}{x} + b}\) (general form)
  • \(\mathrm{f(1) = 6}\) (when x = 1, output is 6)
  • \(\mathrm{f(-1) = 6}\) (when x = -1, output is 6)

• What this gives us:
  • \(\mathrm{f(1) = \frac{a}{1} + b = a + b = 6}\)
  • \(\mathrm{f(-1) = \frac{a}{-1} + b = -a + b = 6}\)

2. INFER that we have a system of equations to solve

• We now have two equations with two unknowns (a and b):
  • \(\mathrm{a + b = 6}\)
  • \(\mathrm{-a + b = 6}\)

• Strategy: Use elimination method since the coefficients of 'a' are opposites

3. SIMPLIFY by solving the system

• Add the equations to eliminate 'a':
  \(\mathrm{(a + b) + (-a + b) = 6 + 6}\)
  \(\mathrm{2b = 12}\)
  \(\mathrm{b = 6}\)

• Substitute \(\mathrm{b = 6}\) back into first equation:
  \(\mathrm{a + 6 = 6}\)
  \(\mathrm{a = 0}\)

4. INFER the final function form

• Since \(\mathrm{a = 0}\) and \(\mathrm{b = 6}\):
  \(\mathrm{f(x) = \frac{0}{x} + 6 = 6}\)

• This means f is actually a constant function!

Answer: B. \(\mathrm{f(x) = 6}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make algebraic errors when solving the system of equations, particularly with the negative coefficient in \(\mathrm{-a + b = 6}\).

A common mistake is incorrectly handling the signs when adding equations or substituting values. For example, they might get confused about whether \(\mathrm{f(-1)}\) gives them \(\mathrm{-a + b}\) or \(\mathrm{a - b}\), leading to an incorrect system. This causes them to calculate wrong values for a and b, potentially leading them to select Choice A (\(\mathrm{f(x) = \frac{3}{x} + 3}\)) if they get \(\mathrm{a = 3, b = 3}\) from calculation errors.

Second Most Common Error:

Poor INFER reasoning: Students may not recognize that when \(\mathrm{a = 0}\), the term \(\mathrm{\frac{a}{x}}\) disappears entirely, making the function a simple constant.

Even if they solve correctly and find \(\mathrm{a = 0, b = 6}\), they might write \(\mathrm{f(x) = \frac{0}{x} + 6}\) and think this is somehow different from \(\mathrm{f(x) = 6}\). They may look for an answer choice that literally contains the "\(\mathrm{\frac{0}{x}}\)" term and get confused when they don't find one, leading to guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can work systematically with function notation and systems of equations, while also recognizing when a seemingly complex rational function simplifies to something much simpler.