Formula, table, graph — one function, three outfits — and the intervals that bound its inputs and outputs.
A function is a rule, and a rule can be written down in more than one way. Spell it out as a formula and you can compute any output you ask for. Lay it out as a table and you can read matched pairs at a glance. Plot those pairs and join them, and the rule becomes a graph you can see all at once. The three views tell one story, and you can pass between them freely. Along the way we name the two sets that fence a function in: the domain of inputs it accepts and the range of outputs it produces — both written with the interval notation we built up in the last stage.
We will follow one friendly function — f(x) = 2x + 1 — through all three views, then meet two more shapes, a parabola and a square-root curve, when we ask exactly which inputs are allowed and which outputs come out. By the end you should be able to start from any one view and rebuild the other two.
The most compact way to pin a rule down is a single line of algebra — a formula. Ours is
f(x) = 2x + 1 (the same as y = 2x + 1).
To find an output, you feed in an input and do the arithmetic. The notation f(2) means "the output of f when the input is 2." Wherever you see x, drop in the number and simplify:
f(0) = 2·0 + 1 = 1, f(1) = 2·1 + 1 = 3, f(2) = 2·2 + 1 = 5.
A formula is exact and tireless: name any input and it hands back exactly one output. That is its great strength — it does not store answers, it computes them on demand.
A formula states the rule in one line. To get the output at an input, substitute the input for x and simplify: f(x) is what comes out when x goes in.
A table is the rule written out as a price-list of matched pairs. One column lists chosen inputs x; the column beside it lists the output y for each, computed once from the formula. You read it straight across: pick a row, and the two numbers in it are a pair (x, y) that the rule produces.
| input x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| output y = 2x + 1 | −5 | −3 | −1 | 1 | 3 | 5 | 7 |
A table is wonderfully concrete — every pair is right there to read — but it only shows the inputs you bothered to list. It is a sample of the rule, not the whole of it. (Notice each output is computed straight from the formula: at x = −2, y = 2·(−2) + 1 = −3.)
Now take each pair from the table and plot it as a point (x, y) on the coordinate plane. The first row gives (−3, −5), the next (−2, −3), and so on. Once they are all down, the pattern leaps out: for y = 2x + 1 the points line up perfectly. Join them and you have the graph — a straight line.
The graph is the view that shows the whole behavior in one glance: how fast the outputs grow, where the rule is positive or negative, what it does far to the left and right. A formula computes, a table samples, a graph reveals the shape.
Because a function gives each input exactly one output, its graph can meet any vertical line at most once. Slide a ruler held straight up-and-down across the picture: if it ever crosses the graph in two places, that single input would have two outputs — so the picture is not the graph of a function. This is the "one output per input" rule, seen with your eyes.
The three forms are not three different things — they are three outfits on the same rule, and they must always agree. Drag the input below and watch a single pair light up at once in the formula, the table, and the graph.
Whichever number you choose, the formula's arithmetic, the highlighted row, and the point on the line carry the very same (x, y). That is what it means for three representations to describe one function.
Not every rule will swallow every number. The domain of a function is the set of all inputs it accepts. We gather those allowed x-values into one set and write it as an interval — exactly the notation from Stage 19 (sets & intervals).
For many rules the domain is every real number. Our line y = 2x + 1 happily accepts any x at all, so its domain is the whole number line, written (−∞, ∞). But two things can shrink a domain:
The domain lives along the x-axis: shade the stretch of x-values the graph is allowed to use, and that shaded band is the domain.
For f(x) = √x: the square root needs a non-negative input, so the smallest allowed x is 0 and there is no upper limit. Domain = [0, ∞) — a closed bracket at 0 (it is allowed), an open ∞.
The range is the partner idea: the set of all outputs the function actually produces as x runs over the whole domain. You read it off the graph by sweeping your eye up the y-axis and noting every height the graph reaches.
Take y = x². Square any real number and you never get below zero — the lowest output is 0 (at x = 0), and the outputs climb forever. So over all real inputs its range is [0, ∞), even though its domain is the whole line (−∞, ∞). Cut the domain down to [−2, 3] and the parabola only reaches from 0 up to 3² = 9, so the range becomes [0, 9].
The domain is read along the x-axis (inputs you put in); the range is read along the y-axis (outputs that come out). They are different sets, and reversing them is the single most common slip — keep "domain = x, range = y" fixed in mind.
Now the round trip. A formula fills a table — just substitute. A table plots as points. Joining the points draws the graph. And a graph read carefully hands the rule back. Every arrow is reversible, and at each stop you are still looking at the same function:
| From this view… | …to this one | How |
|---|---|---|
| formula | table | substitute each x, compute y |
| table | graph | plot each pair (x, y), then join |
| graph | table / formula | read pairs off the picture; spot the pattern |
Pick whichever view answers your question fastest. Need an exact output? Use the formula. Want a few sure pairs? Read the table. Curious where the rule rises, falls, or crosses zero? Look at the graph. Three doors into one room.
Formula, table, and graph are three views of one rule, and you can travel freely between them. The same function carries a domain of inputs (on the x-axis) and a range of outputs (on the y-axis), each written as an interval.
Here is the whole lesson in one breath:
Next, in Stage 20.6, we learn to read a function's behavior straight off its graph — where it climbs and falls, its symmetry, its peaks and valleys.
For f(x) = 2x + 1, fill the table at x = −1, 0, 1, 2.
Substitute each input: f(−1) = 2(−1)+1 = −1; f(0) = 1; f(1) = 3; f(2) = 5. Pairs: (−1, −1), (0, 1), (1, 3), (2, 5).
State the domain of f(x) = 2x + 1.
The line accepts every real number — nothing in 2x + 1 forbids any input. Domain = (−∞, ∞).
State the domain of f(x) = √x, and explain why.
A square root needs a non-negative input, so x ≥ 0. The smallest allowed input is 0 (and it is allowed), with no upper limit. Domain = [0, ∞).
Find the range of y = x² over all real x.
Squaring never gives a negative, and the smallest output, 0, happens at x = 0; outputs climb without bound. Range = [0, ∞).
Find the range of y = x² on the restricted domain [−2, 3].
On [−2, 3] the lowest output is still 0 (at x = 0) and the highest is at the input farthest from 0, namely x = 3: 3² = 9. Range = [0, 9].
Can the graph of a function ever cross a single vertical line twice? Explain.
No. A vertical line fixes one input x; two crossings would mean that input has two outputs, which a function never allows. That is the vertical-line test: at most one crossing per vertical line.
Six questions to lock it in. Tap the answer you think is right.
The big idea is representational fluency: a function is one object that can wear three outfits — a formula, a table, and a graph — and a confident student moves among them at will. We anchor every view to a single friendly rule, f(x) = 2x + 1, so the "same pair, three views" point lands before the abstraction does. The vertical-line test is introduced not as a trick but as the picture of the defining law of functions: one input, one output.
The misconception to watch hardest is swapping domain and range. Keep repeating "domain = inputs = x-axis, range = outputs = y-axis," and have students point to the axis as they answer. Two more slips: forgetting that a formula can restrict the domain (no √ of a negative, no division by 0), and assuming the range is "all of y" when the graph plainly doesn't reach some heights — y = x² never dips below 0, so its range starts at [0, …. The interval brackets carry meaning: square bracket = endpoint included, round bracket = excluded (and ∞ is always round, since you never reach it).
Common Core alignment. This lesson supports 8.F.A.1–3 and F-IF.A.1–2 (function notation; comparing the three representations), F-IF.A.1 and F-IF.B.5 (domain and range, including domain from a context), and 8.F.B / F-IF.C.7 (graphing functions and the vertical-line test). It builds on interval notation (Stage 19) and the coordinate plane and function ideas from Stage 20.1 and 20.4.