1. Language of Functions
1What is a Mapping? What is a Function?
Think of a mapping as an arrow diagram: it takes values from one set (the inputs) and connects them to values in another set (the outputs). That's it — any rule that connects inputs to outputs is a mapping.
A mapping can behave in different ways:
• One-to-one — every single input goes to its own unique output (nobody shares).
• Many-to-one — several different inputs are allowed to land on the output (like x² sending both 3 and −3 to 9).
• One-to-many and many-to-many — these exist too, but there's a problem with them...
The Key Idea
A
function is a mapping where every input produces
exactly one output — no exceptions. If just one input could give you two different outputs, it fails to be a function. That's why only
one-to-one and
many-to-one mappings count as functions — "one-to-many" breaks the rule immediately.
Why does this matter? Because a function needs to be predictable. If you tell a function "put in 5," it needs to give you back one definite answer every single time — not "well, it could be 12 or it could be −3." That reliability is the entire point of a function.
2Function Notation
Functions are usually named with letters like f, g, h. You'll see two main styles of notation, and they mean exactly the same thing:
How a function actually works: whatever sits inside the brackets replaces every x on the right-hand side. If you see f(3), every x becomes a 3. If you see f(a), every x becomes an a. If you see f(x + 3), every x becomes (x + 3) — brackets and all, which is the part that trips people up (see the example below).
Worked Example
Given f(x) = 3x² − 2x + 1, find (a) f(7), and (b) f(x + 3).
(a) Substitute x = 7 into every x:
f(7) = 3(7)² − 2(7) + 1 = 3(49) − 14 + 1 = 147 − 14 + 1
f(7) = 134
(b) Replace every x with the whole bracket (x + 3):
f(x+3) = 3(x+3)² − 2(x+3) + 1
Expand (x+3)² = x² + 6x + 9 first, then distribute the 3 and the −2:
= 3(x² + 6x + 9) − 2x − 6 + 1 = 3x² + 18x + 27 − 2x − 6 + 1
f(x+3) = 3x² + 16x + 22
Common Mistake
When substituting an expression like (x + 3), students often forget to
square the whole bracket — writing x² + 9 instead of (x+3)² = x² + 6x + 9. Always expand brackets carefully, term by term, before simplifying.
Practice Question
A function is defined as g(x) = 2x² + x − 5. Find g(4) and g(2a).
3Domain and Range
These two words confuse people mainly because they sound similar, so let's nail the difference with a simple picture: the domain is the "in" door, the range is the "out" door.
The important connection: the range is entirely the domain. Change what you're allowed to put in, and you change what comes out. This is best seen visually — imagine f(x) = x²:
• If the domain is all real x, then squaring negative and positive numbers both give positive results, so the range is f(x) ≥ 0.
• If you the domain to just x ≥ 0, the mapping becomes one-to-one instead of many-to-one (this trick is often used deliberately, because one-to-one functions are the only ones with proper inverses — more on that soon).
Worked Example
f(x) = (x − 3)² for all values of x. (a) State the range. (b) If the domain changes to x > 5, find the new range.
(a) Any squared value is always ≥ 0, no matter what's inside the brackets, so:
f(x) ≥ 0
(b) Now the smallest allowed input is "just above 5." Substitute the boundary value x = 5 into f to find the boundary of the range: f(5) = (5−3)² = 4. Since x is strictly greater than 5 (not equal), the range is also strict:
f(x) > 4
Handy Technique
To find a range from a domain, imagine (or actually sketch) the
graph of the function. The domain restricts which -values you're allowed to look at; the range is simply whatever -values appear on the curve within that window. Graphing turns an abstract algebra question into a "just read it off the picture" question.
Practice Question
h(x) = 5 − 2x², with domain x ≥ 1. Find the range of h(x).
4The Modulus Function
The modulus function, written with vertical bars like |x|, simply strips away any negative sign — it makes any input positive (or leaves it at zero). It's often called the absolute value, because it just measures "how far from zero," ignoring direction.
Think of the vertical bars like a pair of brackets that "clean up" whatever's inside — anything that comes out negative gets its sign flipped positive on the way out.