Functions, Domain, Range & Graphs

What is a Function?

A function is a rule (or set of instructions) that maps every value of one variable to exactly one value of another variable.

A variable is an unknown quantity — e.g. $x$, $y$ — whose value can change, often depending on the value of some other variable.

A function is usually written as an equation expressing one variable in terms of another:

$y = f(x)$

• $y$ is called the dependent variable — its value depends on $x$.
• $x$ is called the independent variable — you can pick any value for it (within the function's domain).
• The set of values $x$ is allowed to take is called the domain of the function.
• The set of corresponding values $y$ produces is called the range of the function.

You can picture a function as a machine: you feed a value of $x$ in, and it produces exactly one value of $y$ out.

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Worked Examples: Finding Domain and Range

Example 1. $y = x^2 - 10$

This is a function. Since $x$ can be any real number, the domain is:

$x \in \mathbb{R}$

Since $x^2 \geq 0$ for every real $x$, the smallest possible value of $y$ is $-10$ (when $x = 0$). So the range is:

$y \geq -10$

Example 2. $y = \sqrt{x - 1}$

For $y$ to be a real number, the expression under the square root can't be negative:

$x - 1 \geq 0 \quad \Rightarrow \quad x \geq 1$

So the domain is $x \geq 1$, and since a square root is never negative, the range is $y \geq 0$.

Types of Graphs

A graph of a function $f$ is the picture formed by plotting every ordered pair $(x, f(x))$ on a coordinate plane.

1. Straight-line graphs — graphs of $y = mx + c$ (a linear equation). Example: $y = 2x + 1$.

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2. "U-shaped" (even-power) graphs — the highest power of $x$ in the equation is even, e.g. $x^2$, $x^4$.

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A higher even power like $x^4$ flattens near the bottom and rises more steeply at the sides:

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3. "S-shaped" (odd-power) graphs — the highest power of $x$ in the equation is odd, e.g. $x^3$. These curves pass through the origin and flip direction, giving an "S" shape.

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Maxima and Minima

As you trace along a curve, certain points stand out:

• Absolute maximum — the single highest point on the entire graph.
• Relative (local) maximum — a point that's higher than every point immediately around it, even if it isn't the highest point overall.
• Relative (local) minimum — a point that's lower than every point immediately around it.
• Absolute minimum — the single lowest point on the entire graph.
• Point of inflection — a point where the curve changes concavity (switches from curving upward to curving downward, or vice versa).

The curve $y = x^3 - 3x$ shows this clearly — a local maximum at $x = -1$ ($y = 2$) followed by a local minimum at $x = 1$ ($y = -2$):

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