Intuition The one core ideaA function is a reliable machine: you drop in one number, it hands you back exactly one number by a fixed rule. Every "type" in this topic — constant, linear, quadratic, polynomial, rational, radical, piecewise — is just a different shape of rule , and the shape of the rule decides the shape of the picture the machine draws.
Before you can classify functions, you must be fluent in the alphabet they are written in. Below is every symbol and idea the parent note Types of Functions leans on, ordered so each one is built from the ones before it. Nothing is assumed.
Everything starts with two pictures. The figure below shows both: a flat ruler for inputs, a tall ruler for outputs, and how a single dot records one input–output pair.
In the picture, the yellow flat ruler is the input axis, the blue tall ruler is the output axis, and the pink dashed lines trace how the point ( x , y )
Definition The number line andR number line is a straight ruler where every point is a number: negatives to the left, zero in the middle, positives to the right. The symbol R the whole line — every number you can mark on it, including fractions and endless decimals like π
Intuition Why we need the lineA function eats a number and returns a number. To watch it work we need somewhere to put the input and somewhere to put the output — so we use two number lines crossed at a right angle. That crossing is the plane .
Definition The coordinate plane and( x , y ) x y 0 ( x , y ) x right if x x , then go y up if y y . (Zero means don't move along that axis at all.) That single dot is where an input and its output meet — a point on the graph.
Definition The graph of a functionThe graph of f not one dot — it is the whole collection of every point ( x , f ( x )) x {( x , f ( x )) ∣ x is an allowed input } . x x f ( x ) one member of that infinite family.
x y x placeholder for the number you feed in — it can be any value, so we call it the variable . y number that comes out . In the picture, x y
Intuition Why letters instead of numbersWe use a letter because the rule must work for every input at once. Writing "3 × x whatever arrives; writing "3 × 5
The next figure draws the machine idea literally: an input arrow going in, the rule f
Notice in the figure that only one output arrow leaves the box — that single-valued behaviour is what makes f function and not just any rule.
f f ( x ) f name of the rule (like naming the machine "f f ( x ) f x "the output the machine f x The parentheses are not multiplication; they hold the input.
→ ↦ f : R → R f → type of thing in and out. The barbed arrow ↦ x ↦ 3 x specific transformation of one element: "x 3 x
f ( x ) f x Wrong: reading f ( x ) f ⋅ x Fix: f rule . f ( 2 ) f f f ( x ) = 3 x
The parent note's whole job is to say: what does the rule inside those parentheses look like? We'll meet each shape of rule below.
Definition The equals sign "= = (a) "is equal to": a fact you could check, like 2 + 3 = 5 (b) "is defined to be": a naming act, like "let f ( x ) = m x + b inventing the rule, not checking a fact. When in doubt, ask: am I stating a fact or choosing a rule ? Some books write :=
c constant is a number that never changes, written with a fixed letter like c b k f ( x ) = c x c
Intuition Why a letter for a fixed number?So one sentence covers infinitely many cases. "f ( x ) = c c 5 − 2 299 , 792 , 458
We must nail down "power of x before we speak of coefficients or degree, because everything downstream is built from powers. The figure shows how different powers climb at different speeds.
Follow the three chalk curves: the blue line (x x 2 x 3
x n x n multiply x n : x 2 = x ⋅ x x 3 = x ⋅ x ⋅ x n exponent (or power ); here n ≥ 0
Definition The special casex 0 x 0 = 1 any x = 0 — an empty product is 1 0 0 left undefined (conventionally indeterminate): "multiply 0
Intuition Why powers matter for shapeThe highest power present bends the graph the most for big inputs, because a bigger exponent grows faster. x 2 x 3
A coefficient is the number multiplying a power of x 3 x 3 a x 2 a how strongly that piece of x
a n , a n − 1 , … , a 0 subscript is a little index tag attached below a letter: a 0 a 1 a 2 different constants wearing numbered name-tags , one coefficient per power. The tag matches the power it multiplies: a 2 x 2 a 0 x 0 = 1
Now we can assemble the pieces into the family the parent note leans on most.
A polynomial is a finite sum of terms , each a coefficient times a whole-number power of x a n x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 . x x 3 x 2 − 4 x + 7 x 1 x not .
n degree of a polynomial is its largest exponent . Degree 0 1 2 3 x
Definition Naming whole polynomials:p ( x ) q ( x ) f p q whole polynomials so we can talk about them without writing them out. p ( x ) p x p q
The figure shows a straight line with its rise and run marked, so "slope" becomes something you can literally measure with two steps.
In the picture, the blue segment is the sideways run , the pink segment is the upward rise , and the yellow dot on the y b
Definition The change symbolΔ Δ "the change in." Δ x Δ y (new value) − (old value) .
m = Δ x Δ y Slope answers "for each step right, how many steps up?" It is rise over run : m = Δ x Δ y m m m
Dividing Δ y Δ x how far you walked and leaves pure steepness per single step . That single number m
A root (or zero ) is an input x 0 touches or crosses the x . Solving f ( x ) = 0
y b c y x = 0 crosses the y . For a line f ( x ) = m x + b x = 0 f ( 0 ) = b b
Definition The identity symbol≡ ≡ is identically equal to " — equal for every input, not just at one lucky value. "q ( x ) ≡ 0 q all-zeros function : it returns 0 = q ( x ) = 0
= ≡ = is not equal to " (fails at some value); ≡ not identically equal " (not zero everywhere ). In a = 0 don't let this coefficient vanish , or the type collapses (a quadratic with a = 0 q ( x ) ≡ 0 q
Intuition Why division breaksDividing by zero has no answer — there is no number you can multiply by 0 5 rational function q ( x ) p ( x ) p q holes and blow-ups exactly where q ( x ) = 0 asymptotes , discussed under M02.03 Domain and Range .
Definition Rational functionA rational function is one polynomial divided by another:
f ( x ) = q ( x ) p ( x ) , q ( x ) ≡ 0. p ( x ) q ( x ) q domain is every real number except the roots of q — that is, every x q ( x ) = 0 thrown out , because division by zero is undefined. Those excluded inputs are exactly where the graph blows up into vertical asymptotes.
Definition The radical sign x x reverse of squaring: "what non-negative number, squared, gives x " So 9 = 3 3 2 = 9 radical .
n n n x n x n index : 3 8 = 2 2 3 = 8 2
Intuition Why radicals have restricted inputsFor an even index (like the square root), a negative input has no real answer — nothing real, squared, is negative. So a radical function such as f ( x ) = x x ≥ 0
Definition A piecewise functionA piecewise function uses one big brace to glue several rules together, each with the region of x f ( x ) = { rule 1 rule 2 if x is here if x is there x that row's rule. Example:
f ( x ) = { − x x if x < 0 if x ≥ 0 ∣ x ∣
Intuition Why glue rules togetherSome real behaviours change law partway — a taxi fare that jumps after 2 km, a tax rate that rises past a threshold. One formula can't capture the switch, so we hand different inputs to different rules.
Definition Set braces and the builder bar∣ { } set (a collection). { c } c set-builder bar ∣ such that ": { x ∣ x > 0 } x such that x
Definition Interval notation( a , b ) [ a , b ] interval is a stretch of the number line between two endpoints a b Square bracket = endpoint included , round bracket = endpoint excluded :
[ a , b ] a b both ends in .( a , b ) a b both ends out .[ a , b ) a b
We use ∞ round bracket always — ( 0 , ∞ )
Definition Domain and RangeThe domain is every input the machine legally accepts ; the range is every output it can produce . Pictured: domain is the shadow of the graph on the x y M02.03 Domain and Range .
∞ x → ∞ ∞ not a number — it is shorthand for "grows without bound." The arrow → approaches / heads toward ." So x → ∞ x x → a x a near a spot or far out at its ends.
lim lim limit ) is the tool that asks: "as the input approaches some target, what single value does the output settle toward?" The full notation is
lim x → a f ( x ) = L , the limit, as x a f ( x ) L " — meaning "when x a f ( x ) L a lim x → ∞ f ( x ) = L x f ( x ) L end behaviour and horizontal asymptotes .
limit and not just "plug in"
At a spot where the rule breaks — like a rational function's denominator hitting zero, or the "far end" at ∞ cannot simply substitute. The limit lets you talk about where the output is heading even when you can't land exactly on the point. You'll formalise all of this in M03.01 Limits and use it for continuity in M03.02 Continuity .
number line R and plane x y
constant c gives constant function
coefficients a b and subscripts
slope m gives linear function
quadratic when degree is two
polynomial over polynomial gives rational
not equal and not identically zero
end behaviour and asymptotes
Each foundation is a prerequisite brick; the topic note simply stacks them into named shapes.
Cover the right side and test yourself.
What does R The set of all real numbers — every point on the number line.
What is the address ( x , y ) Go x y
What is the graph of a function? The whole set of points ( x , f ( x ))
Is f ( x ) No — it means "the output of rule f x
What does the arrow in f : R → R f
What are the two jobs of "= "Is equal to" (a checkable fact) and "is defined to be" (naming/inventing a rule).
What is a constant c A fixed number that never changes; f ( x ) = c x
What does x n Multiply x n
What is x 0 0 0 x 0 = 1 x = 0 0 0
In a n x n They match — the coefficient's tag equals the power it multiplies.
What is a polynomial? A finite sum of coefficients times whole-number powers of x a n x n + ⋯ + a 0
What do p ( x ) q ( x ) Whole polynomials, so we can stack them as p ( x ) / q ( x )
What is a rational function and its domain? p ( x ) / q ( x ) q ≡ 0 q
What is the degree of a polynomial? Its highest exponent.
What is slope m Rise over run, Δ y /Δ x
What does Δ "The change in" — new value minus old value.
What does ≡ = ≡ =
Why does q ( x ) = 0 Division by zero is undefined, so the function blows up there.
What does x "What non-negative number squared gives x x ≥ 0
How do you read a piecewise brace? Find which region your x
What is a root/zero? An input making the output 0 x
What is the y The output at x = 0 y
Difference between [ a , b ] ( a , b ) Square = endpoint included; round = endpoint excluded.
What does the set-builder bar ∣ "Such that," as in { x ∣ x > 0 }
What does x → ∞ The behaviour of f x
What does lim x → a f ( x ) = L As x a f ( x ) L
Ready? Then head back to the Types topic and watch each rule become a shape.
