A student writes: 'f(3) = 2x + 1 evaluated at x = 3.' Their answer is f(3) = 7. What is wrong with their reasoning, if anything?
ANothing is wrong — f(3) = 7 is correct
BThey used the wrong formula; you must substitute before writing f(3)
Cf(3) means f multiplied by 3, so the answer should be 3f, not 7
DThe domain of f(x) = 2x + 1 does not include x = 3
The reasoning is valid and the answer is correct. f(3) is precisely the question 'what does this function return when the input is 3?' — substituting x = 3 into 2x + 1 gives 7. The common confusion is option C: thinking f(x) means f *times* x. In function notation, f(x) is not multiplication — it means 'the output of function f at input x.' The notation exists specifically to make this input-output relationship explicit.
Question 2 Multiple Choice
A scenario: a phone plan charges $30/month plus $0.10 per text message. Let f(x) = 0.10x + 30, where x is number of texts. A student says 'the domain of this function is all real numbers.' Are they correct?
AYes — as a mathematical function, f(x) = 0.10x + 30 accepts any real number
BNo — the domain must be restricted to positive integers since you can't send a negative or fractional text
CNo — the domain must be restricted to values that make f(x) positive
DYes — but only because the slope is positive
In this context, the student is wrong. The real-world scenario restricts the domain: x must be a non-negative integer (you can't send −5 texts or 2.7 texts). The *mathematical* function f(x) = 0.10x + 30 has all real numbers as its domain, but context can and does restrict it. This distinction — between the pure mathematical domain and the contextually constrained domain — is a key skill in linear functions. The answer is B: the context forces a restriction to whole numbers ≥ 0.
Question 3 True / False
For a non-zero linear function f(x) = mx + b, the range is generally a proper subset of the real numbers.
TTrue
FFalse
Answer: False
False. For f(x) = mx + b with m ≠ 0, the range is *all* real numbers. As x takes every real number value, mx + b hits every real number — the output grows without bound as x increases or decreases. The range is only restricted if m = 0 (a horizontal line), in which case the range is the single value {b}, or if the context restricts the domain. The common misconception is thinking range must be limited the way it is for quadratics or other non-linear functions.
Question 4 True / False
The equation y = 3x + 5 and the equation f(x) = 3x + 5 represent fundamentally the same mathematical object.
TTrue
FFalse
Answer: True
True — they describe the same relationship. Function notation f(x) = 3x + 5 is simply a rewriting of y = 3x + 5 that makes the input-output structure explicit. The variable y is replaced by f(x) to emphasize that 'y is the output of function f when x is the input.' Both have the same slope (3), the same y-intercept (5), and the same graph. The advantage of function notation is practical: f(3) asks a specific question, whereas 'let x = 3 in y = 3x + 5' is more cumbersome and doesn't generalize as cleanly to comparing multiple functions.
Question 5 Short Answer
Why does the slope of a linear function represent a 'constant rate of change,' and why does that constancy matter?
Think about your answer, then reveal below.
Model answer: For every 1-unit increase in input x, the output increases by exactly m — the slope — no matter where on the domain you are. This is what makes the function linear: the rate never accelerates or decelerates. It matters because it allows prediction from any point: if you know the slope and one output, you can compute any other output without needing to re-examine the function.
The constancy distinguishes linear functions from all others. In a quadratic, the rate of change itself changes (it speeds up or slows down). In a linear function, the ratio (change in output)/(change in input) is identical at every point. This means real-world linear models — cost per unit, speed at constant velocity, temperature conversion — allow exact prediction without knowing where you 'started.' The slope is the entire story of how the output responds to the input.