What a function is and how to read f(x); domain and range, on a graph and in a story; the vertical line test; intercepts, zeros, and where a graph rises or falls. Then the linear model: slope as a rate of change, average rate of change on a curve, a line from two points, a table or a sequence, and how to tell linear growth from exponential. Finally a line fitted to data — what its slope and intercept mean in units, and where its predictions stop making sense. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
Put 3 into the rule. 7 comes out. The box is the rule; it does the same thing every time.
f(x) is the output of the rule f at x — never f times x. And f(3) = 7 is the same sentence as the point (3, 7). A function is a rule with exactly one output for every input.
What people get wrong
⚠️People often think…
f(x) means f times x.
f is the name of a rule, not a number, so there is nothing to multiply. f(x) is what the rule gives back when x goes in. The parentheses mean “of,” not “times.”
Read it out loud as “f of x.”
⚠️People often think…
{(1, 4), (2, 4)} is not a function, because 4 repeats.
Repeated OUTPUTS are fine — two inputs are allowed to give the same answer. What breaks a function is one INPUT with two different outputs, because then the rule cannot decide. Check the x column.
Check the inputs for repeats. Never the outputs.
⚠️People often think…
Domain and range are more or less interchangeable.
Domain is the inputs — the x's, read left to right. Range is the outputs — the y's, read bottom to top. Swap them on a word problem and the answer stops meaning anything.
Domain sweeps across. Range sweeps up.
⚠️People often think…
A slope of −1.5 means the price drops a dollar fifty a year.
Read the axis label first. If the y-axis is in thousands of dollars, a slope of −1.5 is $1,500 a year. The number on the graph is never the whole story — the units are.
Slope carries units: y-units per x-unit.
Find b, find m, say what they mean
1Watch one
f(x) = 2x + 1. Find f(3).
f(3) means: put 3 in for x.
2(3) + 1.
6 + 1 = 7.
f(3) = 7 — which is the point (3, 7).
2Do one with me
Still f(x) = 2x + 1. Read the two numbers that describe the whole line.
f(0) =
That point is called the
Every time x goes up 1, y goes up
💬One sentence, then you move on
Why does putting zero in for x give you b?
3Try one
Is {(1, 4), (2, 4), (3, 9)} a function? Write yes or no.
I want a hint first
List only the inputs: 1, 2, 3. Did any of them show up twice with different answers?
💬Last one — then you're done here
Why is a repeated output allowed but a repeated input not?
Where this goes
Where this lives
A pay scale, a phone plan, the depreciation on a car — anything with a starting amount and a steady rate is this, and the axis units decide what the numbers mean.
What this feeds
Next unit is systems — two of these lines at once, and the one point that satisfies both.
Name one rule from your life that takes an input and gives exactly one output.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🧾 Reading a linear model — the routine
Find the two numbers: m multiplies x (the rate), b stands alone (the start) → attach units: m is “output units per input unit,” b is just output units → say each one as a sentence about the situation → check with one input: put x = 1 in and confirm the output is b + m → decide the sensible domain (can x be negative? a fraction? bigger than what exists?).
Rate, start, units, sentence, check.
💡 The whole unit in one idea
A function is a dependable machine: one input, one output, every time. A LINEAR function is the simplest machine of all — it changes by the same amount for every unit of input — so two numbers, the rate and the starting value, describe it completely, whether it arrives as a formula, a table, a graph, a sequence or a line drawn through a cloud of data. Everything in this unit is either checking that a rule really is a function, or finding those two numbers and saying what they mean.
⚠️ Traps the test loves
Reading f(x) as “f times x.” It is the output of f at x. f(3) = 7 is the point (3, 7).
Calling {(1, 4), (2, 4)} “not a function” because 4 repeats. Shared OUTPUTS are fine; shared INPUTS are not.
Swapping domain and range. Domain is the x’s (sweep left-right); range is the y’s (sweep bottom-top).
Using a point’s y as b when the point is not on the y-axis. Substitute and solve for b.
Forgetting to divide by the run: the rise alone is not the slope.
Ignoring axis units: a slope of −1.5 with prices in thousands is $1,500 a year, not $1.50.
📋 One linear function, five disguises — the chart
It arrives as…
The rate (m) is…
The start (b) is…
a formula f(x) = mx + b
the coefficient of x
the constant term
a table with x going up by 1
the difference between consecutive y’s
y when x = 0 (or count back to it)
a graph
rise over run between two lattice points
where it crosses the y-axis
two points
(y₂ − y₁)/(x₂ − x₁)
solve y = mx + b with one point
a sequence 4, 7, 10, …
the common difference
f(1) − m, so f(n) = 3n + 1
a story (fee plus rate)
the “per” number, with units
the one-time amount, with units
🎯 How the test will ask
A set of pairs, a table or a graph — “is it a function?” Look for one input with two outputs; on a graph, the vertical line test.
f(x) = 3x − 1 — “find f(4)” and “find x when f(x) = 11.” One evaluates, one solves.
A graph — “state the domain, range and intercepts,” or “where is it increasing?” Read x-values for domain and intervals, y-values for range.
Two points or a table — “write the linear function.” Slope first, then solve for b, then check the other point.
A story with a fee and a rate — “what do the slope and y-intercept mean?” Answer in sentences with units.
Two tables — “linear or exponential?” Subtract neighbors for a constant difference; divide for a constant ratio.
A scatterplot with a line of fit — “interpret the slope, predict a value, and say when the prediction fails.”
✅ Before the test, can you…
Decide whether a set of pairs, a table, or a graph is a function, and say why?
Evaluate f(a) and solve f(x) = k from a formula, a table, or a graph?
Read the domain, range, intercepts and intervals of increase from a graph, and give a sensible domain in a story?
Find the slope from two points, a table or a graph, and compute an average rate of change for a curve?
Write f(x) = mx + b from two points, a sequence, or a story, and check it?
Tell linear growth from exponential growth in a table or a description?
Interpret the slope and intercept of a line of fit with the right units, and explain why extrapolating far from the data is risky?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a mathematician: read the domain, range and intercepts off a graph, plot a study-time table on a coordinate plane with your fingertip and name its slope, argue what a fitted line’s slope really says about a used car, then decide which rules are functions and which of those are linear. Every activity checks itself, and hints are free.
Your practice never leaves this device. There is no account and no sign-in. Your work is saved in this browser only, and you can erase it whenever you want.
Your practice record — saved on this device
This is your record of the module on screen — it stays here and goes nowhere. Independent means you got it right on the first tap; supported means you got it after the explain-and-retry, or marked ‘I had it’ on a revealed answer. Both count, and neither is a grade. If your teacher asks, copy the row or show them this screen.
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