AP PRECALCULUS — PREREQUISITE REVIEW
Representations of Functions
Notes — Prerequisite Topic 9
💡 Learning Objectives By the end of this lesson you will be able to:
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1. What is a Function?
A function is a rule that assigns to every input exactly one output. The input is usually called x (the independent variable) and the output is usually called y (the dependent variable). We write f(x) for the output produced by the input x; this is read as ‘f of x.’
⚠️ The ‘exactly one’ condition matters If a single input can be matched with two different outputs, the rule is not a function. For example, the rule that assigns to a positive integer ‘every positive divisor of it’ is not a function, because 6 would be matched with 1, 2, 3, and 6. |
2. Four Ways to Represent a Function
The same function can often be expressed in four equivalent ways. A skilled mathematician moves fluidly between them and picks whichever form is most useful for the question at hand.
Representation | What it looks like | Best used for |
|---|---|---|
Analytical (formula) | f(x) = 2x + 1 | computing exact values, algebraic manipulation |
Numerical (table) | input–output pairs | seeing patterns, fitting models to data |
Graphical | a curve in the plane | spotting behaviour, features, and trends |
Verbal (words) | a sentence or description | modelling real-world situations |
The same linear function y = 2x + 1 in graphical and numerical form. Every table value sits on the graph; every graph point corresponds to exactly one table entry.
2.1 From a formula to a table
Plug chosen inputs into the formula; record each output.
2.2 From a table to a formula
Look for a pattern in the output column. Is the first difference constant? The function is linear. Is the ratio of consecutive outputs constant? It is exponential. Is the second difference constant? It is quadratic.
📘 Example — From table to formula Given the table: x: 0, 1, 2, 3, 4 y: 5, 2, −1, −4, −7 First differences: −3, −3, −3, −3 → constant, so the function is linear with slope −3. The y-intercept is 5, so f(x) = −3x + 5. |
2.3 From verbal to analytical
Translating words into formulas is the single most important modeling skill in precalculus. Name each quantity with a variable, then write equations that match the sentences.
📘 Example — From words to a formula ‘A taxi charges a $3 fee to enter the cab plus $2.50 per mile travelled.’ Let C = cost in dollars for a ride of m miles. Then C(m) = 3 + 2.50m. |
3. The Vertical Line Test
A graph in the xy-plane represents y as a function of x if and only if every vertical line crosses the graph at most once.
Left: a parabola is a function because every vertical line meets it once. Right: a circle is not a function because a vertical line can meet it twice — that single x-value would correspond to two y-values.
4. Domain and Range
💡 Two essential sets
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When a function is given by a formula alone, its natural domain is all real numbers for which the formula makes sense. Exclude any input that would:
- make a denominator zero
- force a square root (or any even-indexed radical) to have a negative radicand
- take the logarithm of a non-positive number
📘 Example — Finding a natural domain f(x) = √(x − 4): requires x − 4 ≥ 0, so domain is [4, ∞). g(x) = 1 / (x² − 9): denominator zero at x = ±3, so domain is {x ≠ 3, −3}. h(x) = x² + 1: every real number works, so domain is (−∞, ∞). |
5. Function Notation in Use
Function notation lets you write carefully and unambiguously. Some examples of what it expresses:
- f(5) means the output of f at the input 5
- f(a + 1) means substitute (a + 1) everywhere x appears in the formula
- f(x + h) − f(x) is the change in output when the input moves from x to x + h
📘 Example — Working with function notation Let f(x) = x² − 3x + 2. f(4) = 16 − 12 + 2 = 6. f(a + 1) = (a + 1)² − 3(a + 1) + 2 = a² + 2a + 1 − 3a − 3 + 2 = a² − a. f(x + h) − f(x) = (x + h)² − 3(x + h) + 2 − (x² − 3x + 2) = 2xh + h² − 3h = h(2x + h − 3). |
6. Interpreting Functions in Context
When a function models a real-world situation, every symbol carries meaning. The input, the output, the slope, the intercepts — all correspond to something concrete.
📘 Example — Interpreting a cost function For C(m) = 3 + 2.50m (the taxi cost): • C(0) = 3 is the minimum charge — just getting in the cab. • The slope 2.50 is the cost per additional mile. • C(10) = 28 is the cost of a 10-mile ride. • The domain m ≥ 0 reflects that miles cannot be negative. |
7. Summary
- Each input of a function has exactly one output; vertical-line test confirms this on a graph
- Move fluidly among graph, table, formula, and words — each clarifies different questions
- Natural domain is the largest set of inputs for which the formula makes sense
- Function notation is precise shorthand; f(a + h) means substitute (a + h), not ‘multiply by the outside’
- In contextual problems, slopes and intercepts always have real-world meaning worth stating