1. TRANSLATE the function notation
- Given information:
- Function \(\mathrm{f(x) = 39}\)
- What this tells us: For any input value x, the function output is 39
2. INFER what type of function this represents
- This is a constant function because the output never changes
- No matter what x value we choose, \(\mathrm{f(x)}\) will always equal 39
- So \(\mathrm{f(0) = 39}\), \(\mathrm{f(1) = 39}\), \(\mathrm{f(2) = 39}\), and \(\mathrm{f(any\ number) = 39}\)
3. TRANSLATE each table to check which matches our function
- Choice A: All \(\mathrm{f(x)}\) values are 0 → doesn't match \(\mathrm{f(x) = 39}\)
- Choice B: All \(\mathrm{f(x)}\) values are 39 → matches perfectly!
- Choice C: \(\mathrm{f(x)}\) values are 0, 39, 78 → these change, so not constant
- Choice D: \(\mathrm{f(x)}\) values are 39, 0, -39 → these also change, so not constant
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret \(\mathrm{f(x) = 39}\) as meaning "x equals 39" or think it represents a more complex linear relationship.
They might look for a pattern in the x-values (0, 1, 2) and expect the \(\mathrm{f(x)}\) values to follow some increasing or decreasing pattern. This confusion about function notation leads them to dismiss the constant table in Choice B as "too simple" and instead select Choice C (with values 0, 39, 78) because it shows x and \(\mathrm{f(x)}\) both increasing.
Second Most Common Error:
Poor INFER reasoning: Students recognize that \(\mathrm{f(x) = 39}\) means the output is 39, but fail to understand that this applies to ALL possible x values.
They might think that \(\mathrm{f(x) = 39}\) only applies to one specific x value and look for a table where 39 appears at least once. This leads them to select Choice D (with values 39, 0, -39) because it contains the value 39, even though it also contains other values.
The Bottom Line:
Understanding function notation is crucial - \(\mathrm{f(x) = 39}\) means "39 is the output for any input," not "39 is somehow related to the input." Constant functions might seem "too easy," but that's exactly what they are!
