The function \(\mathrm{f(w) = 6w^2}\) gives the area of a rectangle, in square feet (ft²), if its width is w...

1. TRANSLATE the function notation

• Given information:
  • \(\mathrm{f(w) = 6w^2}\) represents area of rectangle
  • \(\mathrm{w}\) = width in feet
  • \(\mathrm{f(14) = 1{,}176}\)

• What \(\mathrm{f(14) = 1{,}176}\) tells us:
  • Input: \(\mathrm{w = 14}\) (width = 14 ft)
  • Output: \(\mathrm{f(14) = 1{,}176}\) (area = 1,176 ft²)

2. INFER the contextual meaning

• Since \(\mathrm{f(w)}\) gives area when width is \(\mathrm{w}\):
  • \(\mathrm{f(14)}\) gives area when width is 14 ft
  • Therefore: area = 1,176 ft² when width = 14 ft

3. Match with answer choices

• Look for: "width = 14 ft" AND "area = 1,176 ft²"
• Choice A matches exactly: "If the width of the rectangle is 14 ft, then the area of the rectangle is 1,176 ft²"

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the input and output values in function notation.

They might think \(\mathrm{f(14) = 1{,}176}\) means "f of 1,176 equals 14" instead of "f of 14 equals 1,176." This backwards interpretation leads them to think the width is 1,176 ft and some other value is 14.

This may lead them to select Choice C (width = 1,176 ft, length = 14 ft) or Choice D (width = 1,176 ft, area = 14 ft²).

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly identify that width = 14 ft but then confuse what the output value 1,176 represents.

Since the problem mentions "length is 6 times the width," they might think 1,176 represents the length instead of recognizing that \(\mathrm{f(w)}\) specifically represents area.

This may lead them to select Choice B (width = 14 ft, length = 1,176 ft).

The Bottom Line:

Function interpretation problems require careful attention to what the input and output represent in context. The key is matching the function notation structure (input → output) with the real-world meaning.