The function g is linear and \(\mathrm{g(0) = 0}\). If \(\mathrm{g(7) = -84}\), what is \(\mathrm{g(1)}\)?

1. TRANSLATE the problem information

• Given information:
  • g is a linear function
  • \(\mathrm{g(0) = 0}\) (function passes through origin)
  • \(\mathrm{g(7) = -84}\)

2. INFER the function form

Since g is linear and passes through the origin (\(\mathrm{g(0) = 0}\)), we know:

• Linear functions have form \(\mathrm{g(x) = mx + b}\)
• Since \(\mathrm{g(0) = 0}\), we have: \(\mathrm{0 = m(0) + b}\), so \(\mathrm{b = 0}\)
• Therefore: \(\mathrm{g(x) = kx}\) for some constant k

3. INFER how to find the constant

We can use the given point \(\mathrm{g(7) = -84}\) to find k:

• Substitute into \(\mathrm{g(x) = kx}\): \(\mathrm{g(7) = k(7) = 7k}\)
• Set equal to given value: \(\mathrm{7k = -84}\)

4. SIMPLIFY to find k

\(\mathrm{7k = -84}\)
\(\mathrm{k = -84/7 = -12}\)

5. SIMPLIFY to find the answer

Now we have \(\mathrm{g(x) = -12x}\)
Therefore: \(\mathrm{g(1) = -12(1) = -12}\)

Answer: B) -12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may not recognize that "\(\mathrm{g(0) = 0}\)" means the function passes through the origin, leading them to use the general linear form \(\mathrm{g(x) = mx + b}\) instead of the simplified form \(\mathrm{g(x) = kx}\).

This confusion about the function form can cause them to set up incorrect equations or get overwhelmed by unnecessary complexity, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that \(\mathrm{g(x) = kx}\) but make arithmetic errors when calculating \(\mathrm{k = -84/7}\), perhaps getting \(\mathrm{k = 12}\) instead of \(\mathrm{k = -12}\).

This leads them to calculate \(\mathrm{g(1) = 12}\), making them select Choice D (84) if they think the answer should be positive, or get confused about the sign.

The Bottom Line:

This problem tests whether students can recognize that a linear function through the origin has a special simplified form, then use given information systematically to find the constant. The key insight is translating the conditions into the right mathematical representation.