For the linear function f, \(\mathrm{f(0) = 17}\) and \(\mathrm{f(1) = 17}\). Which equation defines f?

1. TRANSLATE the given information

• Given information:
  • f is a linear function
  • \(\mathrm{f(0) = 17}\) (when input is 0, output is 17)
  • \(\mathrm{f(1) = 17}\) (when input is 1, output is 17)

• What this tells us: We have two coordinate points \(\mathrm{(0, 17)}\) and \(\mathrm{(1, 17)}\)

2. INFER the solution strategy

• Since f is linear, it has the form \(\mathrm{f(x) = mx + b}\)
• We need to find the slope \(\mathrm{m}\) and y-intercept \(\mathrm{b}\)
• The y-intercept \(\mathrm{b}\) is the value when \(\mathrm{x = 0}\), so \(\mathrm{b = 17}\)
• We can use the slope formula to find \(\mathrm{m}\)

3. SIMPLIFY to find the slope

• Using slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• With points \(\mathrm{(0, 17)}\) and \(\mathrm{(1, 17)}\):

\(\mathrm{m = \frac{17 - 17}{1 - 0}}\)

\(\mathrm{m = \frac{0}{1}}\)

\(\mathrm{m = 0}\)

4. Form the final equation

• Substituting \(\mathrm{m = 0}\) and \(\mathrm{b = 17}\) into \(\mathrm{f(x) = mx + b}\):
• \(\mathrm{f(x) = 0x + 17 = 17}\)

Answer: C. \(\mathrm{f(x) = 17}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that when both function values are the same (\(\mathrm{f(0) = 17}\) and \(\mathrm{f(1) = 17}\)), this indicates a horizontal line with zero slope.

Instead, they might think the function somehow involves both the numbers 17 and 1, leading to confused calculations like thinking the function should be \(\mathrm{f(x) = 17x}\) or trying to combine 17 and 1 in some way. This may lead them to select Choice D (\(\mathrm{f(x) = 34}\)) by adding the values, or get confused and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the slope formula but make an arithmetic error, thinking that \(\mathrm{\frac{17-17}{1-0}}\) equals 1 instead of 0, possibly because they focus on the denominator being 1.

This leads them to believe the slope is 1, so they conclude \(\mathrm{f(x) = 1x + 17 = x + 17}\). When checking against the answer choices, this doesn't match exactly, causing them to select Choice B (\(\mathrm{f(x) = 1}\)) as the closest option.

The Bottom Line:

This problem tests whether students understand that a linear function with the same output for different inputs represents a constant (horizontal) function, requiring them to recognize the geometric meaning of zero slope.