2.1 Function Notation and Basic Concepts

Function Notation and Terminology

Functions are the core building blocks of Algebra II. A function connects each input to exactly one output, and function notation gives you a compact way to express that relationship. Understanding how to read, evaluate, and classify functions is essential for nearly everything else in this course.

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Functions and Function Notation

A function is a relation that assigns each element $x$ in a set $A$ to exactly one element, called $f(x)$, in a set $B$. The key word is exactly one: every input gets one and only one output.

Function notation is written as $f(x)$, where $f$ names the function and $x$ is the input variable. You read $f(x)$ as "f of x." It does not mean $f$ times $x$.

To evaluate a function, substitute the input value wherever you see $x$:

• If $f(x) = 2x + 1$, then $f(3) = 2(3) + 1 = 7$
• If $g(x) = x^2 - 4x$, then $g(-1) = (-1)^2 - 4(-1) = 1 + 4 = 5$

You can also evaluate functions at expressions, not just numbers. For instance, $f(a + 1) = 2(a + 1) + 1 = 2a + 3$.

Domain, Codomain, and Range

• The domain is set $A$, the set of all allowable inputs.
• The codomain is set $B$, the set of all potential outputs.
• The range is the set of outputs the function actually produces. The range is always a subset of the codomain (it can equal the codomain, but it doesn't have to).

The vertical line test determines whether a graph represents a function:

• If any vertical line intersects the graph more than once, the relation is not a function, because that would mean one input maps to multiple outputs.
• A circle fails the vertical line test because a vertical line through its interior hits the curve at two points.

Domain and Range of Functions

Determining Domain and Range

The domain is the set of all $x$-values for which the function is defined. The range is the set of all $y$-values the function actually outputs.

You can find domain and range from different representations:

• From a graph: The domain is the set of all $x$-values that have a point on the graph. The range is the set of all $y$-values that appear on the graph. Look at how far left/right and up/down the graph extends.
• From an equation: Think about what inputs could cause problems (more on this below).
• From a table: The domain is the set of $x$-values listed; the range is the set of corresponding $y$-values.

Domain Restrictions

Certain operations create values of $x$ that you must exclude from the domain:

• Division by zero: The denominator of a rational function cannot equal zero. For $f(x) = \frac{1}{x - 2}$, set $x - 2 = 0$ to find that $x = 2$ must be excluded. The domain is all real numbers except $x = 2$.
• Square roots of negatives: The expression under an even-index radical must be greater than or equal to zero. For $f(x) = \sqrt{x + 1}$, set $x + 1 \geq 0$ to get $x \geq -1$. The domain is $[-1, \infty)$.

To find domain restrictions systematically:

1. Identify any denominators and set each one $\neq 0$. Solve to find excluded values.
2. Identify any even-index radicals and set each radicand $\geq 0$. Solve to find the allowed interval.
3. If both types appear in the same function, apply both restrictions and take the intersection.

Function Classifications

One-to-One Functions (Injective)

A function is one-to-one (injective) if no two different inputs produce the same output. Formally, whenever $a \neq b$, then $f(a) \neq f(b)$.

The horizontal line test checks this graphically:

• If any horizontal line intersects the graph more than once, the function is not one-to-one.
• $f(x) = x^2$ fails because the horizontal line $y = 4$ hits the graph at both $x = -2$ and $x = 2$.
• $f(x) = x^3$ passes because every horizontal line crosses the graph at most once.

One-to-one functions matter because only one-to-one functions have inverses that are also functions.

Onto Functions (Surjective)

A function is onto (surjective) if every element in the codomain is actually hit by some input. In other words, the range equals the codomain with nothing left over.

To check whether a function is onto, you need to know what the codomain is. For example, $f(x) = x^2$ with codomain $\mathbb{R}$ (all real numbers) is not onto because negative numbers in the codomain are never produced. But if you restrict the codomain to $[0, \infty)$, then $f(x) = x^2$ is onto.

Bijective Functions

A function is bijective if it is both one-to-one and onto. Every element in the codomain is paired with exactly one element in the domain. Bijective functions are the only functions that have inverses which are themselves bijective. You'll rely on this concept heavily when you study inverse functions later in the course.

Functions vs Relations

Defining Relations

A relation is any set of ordered pairs $(x, y)$ that connects elements from one set to another. Every function is a relation, but not every relation is a function. Relations can be represented as graphs, equations, tables, or mapping diagrams.

Determining if a Relation is a Function

For a relation to qualify as a function, each $x$-value must pair with exactly one $y$-value. Here's how to check:

• From a graph: Apply the vertical line test. If every vertical line crosses the graph at most once, it's a function.
• From a set of ordered pairs: Check that no $x$-value appears more than once with a different $y$-value. The relation $\{(1,2),\; (2,3),\; (1,4)\}$ is not a function because $x = 1$ maps to both $2$ and $4$.
• From a table: Same idea as ordered pairs. Scan the input column for repeated $x$-values with different outputs.

Note that two different $x$-values can map to the same $y$-value and still be a function. The set $\{(1,3),\; (2,3)\}$ is a valid function; it just isn't one-to-one.