The definition
Feed it a number, it gives back exactly one number. That "exactly one" is the whole definition. A rule that answers maybe 2, maybe −2 is not a function, it is a relation, and everything on this page would collapse without the difference.
The machine
Domain & image
The domain is the set of legal inputs. The range is the set of values actually reached. Two rules generate almost every restriction you will meet: never divide by zero, never take an even root of a negative number.
Restrict the domain
- Fraction
- $\dfrac{1}{x-2}$ needs $x\neq 2$. The graph answers with a vertical asymptote.
- Even root
- $\sqrt{x+3}$ needs $x\geqslant -3$. The curve simply starts there.
- Logarithm
- $\log_b x$ needs $x>0$, strictly.
- Tangent
- $\tan x$ needs $x\neq\frac{\pi}{2}+\pi k$, one hole every half turn.
The graph
Every point on the curve is one answer from the machine. Drag the tracer and watch the pair move together: the horizontal distance is what you put in, the vertical distance is what came out.
The tracer
The kitchen
Every transformation you will ever need sits in one line. Choose a base shape, then four knobs do the rest. Nothing here is memorised, it is tasted: move a slider and watch which way the curve answers.
Ingredient
Seasoning
Table de valeurs
- a
- stretches vertically. $|a|>1$ taller, $|a|<1$ flatter, $a<0$ flips it upside down.
- b
- squeezes horizontally by the factor $1/b$. Bigger $b$, narrower shape. $b<0$ mirrors left to right.
- h
- slides right by $h$. The sign lies: $f(x-3)$ moves right three.
- k
- slides up by $k$. This one is honest.
Épreuves the page checks by itself—
Composition
$(f\circ g)(x)=f(g(x))$ means: run $g$ first, hand the result to $f$. Swap the order and you usually get a different function, which is the point of the exercise.
First machine · g
Second machine · f
The inverse
The inverse undoes the function: $f^{-1}(f(x))=x$. It exists only when no output is ever produced twice, because otherwise running backwards would have to guess. Geometrically the inverse is the same curve reflected in the line $y=x$.
Choose a machine
Parity & period
Test a shape
- Even
- mirror across the vertical axis. $x^2$, $\cos x$, $|x|$.
- Odd
- half turn around the origin. $x$, $x^3$, $\sin x$, $\tan x$.
- Periodic
- $f(x+T)=f(x)$ for every $x$. $\sin$ and $\cos$ repeat every $2\pi$, $\tan$ every $\pi$.
The catalogue
The quadratic
Coefficients
The circle
Walk around a circle of radius 1. Your height above the centre is $\sin\theta$, your distance sideways is $\cos\theta$. Unroll that walk along a horizontal axis and the wave appears. Nothing else is happening.
The angle
- Radians
- the angle measured as arc length on the unit circle. A full turn is $2\pi$, a straight angle is $\pi$, a right angle is $\frac{\pi}{2}$. Degrees to radians: multiply by $\frac{\pi}{180}$.
- Signs
- quadrant I all positive, II only sine, III only tangent, IV only cosine.
- Exact values
- $\sin\frac{\pi}{6}=\frac12$, $\sin\frac{\pi}{4}=\frac{\sqrt2}{2}$, $\sin\frac{\pi}{3}=\frac{\sqrt3}{2}$, and cosine reads the same list backwards.
The wave
The identities
Pick one
Reading a graph
- Zeros
- where it meets the horizontal axis. These are the solutions of $f(x)=0$.
- y-intercept
- the single value $f(0)$, if $0$ is in the domain.
- Sign
- above the axis $f>0$, below it $f<0$. Sign can only change at a zero or at a hole.
- Growth
- as $x$ moves right, does the height rise or fall? Name the intervals.
- Extremes
- the highest and lowest points reached, locally or over the whole domain.
- Asymptotes
- lines the curve hugs without touching. Vertical from zeros of a denominator, horizontal from the far behaviour.
- Ends
- what happens far left and far right.
- Symmetry
- even, odd, periodic, or none of the three.
Interrogate this curve
The slope
Pin two points on the curve and draw the line through them. Now slide them together: the secant settles into the tangent, and its slope $f'(x_0)$ is the number the whole of calculus is built on. Squeeze $\Delta$ to almost nothing and watch the two slopes agree.