Functions & Their Graphs

In variables and the function machine we met a mysterious machine: drop a number in, and it hands another number back according to a hidden rule. Time to give the machine its proper name — a function — and take one step further: plot its inputs and outputs on graph paper, and the machine's temperament appears as a curve.

A machine with input and output

A function is exactly such a machine:

Input (written $x$ ): exactly one number goes in;
Rule: fixed forever, say "double it, then add 1";
Output: whatever the rule produces.

Once the rule is set, every input has exactly one output — one input may never produce two answers. That is the iron law of functions.

f(x): naming the rule

"Double it, then add 1" is written $f(x) = 2x + 1$ — read "f of x", meaning: raw material $x$ , processed by rule $f$ . Asking for $f(3)$ asks "what comes out if we feed in 3?" — $f(3) = 2 \times 3 + 1 = 7$ . Likewise $f(10) = 21$ and $f(0) = 1$ .

From table to graph

Make an input–output table for $y = x^2$ :

$x$	−2	−1	0	1	2
$y$	4	1	0	1	4

Plot each pair $(x, y)$ as a point, connect them with a smooth curve, and a U-shaped parabola appears. Different rules grow different shapes: rules like $y = 2x + 1$ draw straight lines, rules like $y = x^2$ bend into parabolas, and more complicated rules make curves that wave up and down.

Symmetry hiding in the table

$-1$ and $1$ both give 1; $-2$ and $2$ both give 4 — squaring erases the minus sign. That is why the parabola is mirror-symmetric, and why it is not a line: the output runs faster than the input. Double the input, and the output quadruples.

Reading a graph

The graph is the machine's personality sheet:

Rising and falling: reading left to right, a climbing curve means "bigger input, bigger output"; a sinking one means the reverse;
Intercepts: where the curve crosses the $y$ -axis (the output when $x = 0$ ), and where it lands on the $x$ -axis (the input whose output is exactly 0);
Domain and range, in one breath: the domain is every input the machine accepts, the range is every output it could possibly produce.

InteractiveFunction Grapher

Slope m1Intercept b2

y = x + 2

y-intercept

(0, 2)

x-intercept

(-2, 0)

Drag $m$ and $b$ : $m$ controls the slope, $b$ the starting point. The line $y = 2x + 1$ is what $m = 2$ , $b = 1$ looks like, crossing the $y$ -axis exactly at 1. All further line tricks wait in straight line equations.

Check yourself

Quick quiz

1. If f(x)=2x+1, what is f(3)?
2. If y=x², what is y when x=−3?
3. Where does the line y=2x+1 cross the y-axis?