In the xy-plane, the graph of the function g crosses the x-axis at a point with coordinates \(\mathrm{(a, 0)}\). Which...

1. TRANSLATE the problem information

• Given information:
  • The graph of function g crosses the x-axis at point \(\mathrm{(a, 0)}\)
• What this tells us: We have a specific point on the function's graph

2. INFER the relationship between graph points and function notation

• Key insight: Any point \(\mathrm{(x, y)}\) on a function's graph means \(\mathrm{g(x) = y}\)
• For our point \(\mathrm{(a, 0)}\): the input is a and the output is 0
• Therefore: \(\mathrm{g(a) = 0}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the roles of coordinates and mix up input/output relationships.

They see the point \(\mathrm{(a, 0)}\) and incorrectly think "\(\mathrm{g(0) = a}\)" because they associate the first coordinate with the function input being 0. This reverses the actual relationship and leads them to select Choice B (\(\mathrm{g(0) = a}\)).

Second Most Common Error:

Conceptual confusion about function notation: Students don't fully grasp that coordinates directly translate to input-output pairs.

They might think that since the point involves both a and 0, the function relationship should be \(\mathrm{g(a) = a}\), missing that the y-coordinate (which is 0) determines the function's output. This may lead them to select Choice C (\(\mathrm{g(a) = a}\)).

The Bottom Line:

This problem tests the fundamental bridge between visual graph interpretation and algebraic function notation. Success requires precisely translating coordinate information into function relationships.