3.1 Functions and Function Notation

Functions and Function Notation

A function is a rule that assigns exactly one output to each input. This idea shows up everywhere in algebra and beyond, so getting comfortable with function notation and how to work with functions is essential for the rest of the course.

This section covers what makes something a function, how to evaluate and analyze functions, and the most common types you'll encounter.

Functions and Function Notation

Functions vs. General Relations

A relation is any set of ordered pairs connecting inputs to outputs. A function is a specific type of relation where each input produces exactly one output.

Think of it this way: if you plug in $x = 3$ and get back both $5$ and $-2$, that's a relation but not a function. A function demands a single answer for every input.

• Every function is a relation, but not every relation is a function.
• No two ordered pairs in a function can share the same input (first coordinate) with different outputs (second coordinates).
• Example: $\{(1, 4), (2, 7), (3, 4)\}$ is a function because each input appears once. $\{(1, 4), (1, 7)\}$ is not a function because the input $1$ maps to two different outputs.

Evaluation of Function Inputs

Function notation like $f(x)$ tells you the name of the function ($f$) and the input variable ($x$). The expression $f(3)$ means "plug $3$ in for $x$ and simplify."

Here's how to evaluate a function step by step:

1. Start with the function rule. Say $f(x) = 2x + 1$.
2. Replace every $x$ in the expression with the given input value.
3. Simplify the result.

For $f(3)$: substitute to get $f(3) = 2(3) + 1 = 7$.

You can also evaluate at expressions, not just numbers. If you're asked for $f(a + 1)$, replace $x$ with $(a + 1)$:

$f(a + 1) = 2(a + 1) + 1 = 2a + 2 + 1 = 2a + 3$

In applied problems, the output often represents something concrete like cost, temperature, or distance, so pay attention to context.

One-to-One Function Analysis

A one-to-one function takes the uniqueness requirement a step further: not only does each input give exactly one output, but each output comes from exactly one input. No two different inputs produce the same output.

• $f(x) = 2x + 1$ is one-to-one because different inputs always give different outputs.
• $f(x) = x^2$ is not one-to-one because $f(3) = 9$ and $f(-3) = 9$.

To test graphically, use the horizontal line test: if any horizontal line crosses the graph more than once, the function is not one-to-one.

Why does this matter? One-to-one functions are the only functions that have inverses which are also functions. You'll rely on this property when you study inverse functions later in the course.

Vertical Line Test for Functions

The vertical line test is the graphical way to check whether a relation is a function.

1. Look at (or imagine drawing) vertical lines across the graph at various $x$-values.
2. If every vertical line hits the graph at most once, the relation is a function.
3. If any vertical line hits the graph in two or more places, it's not a function (that $x$-value would have multiple outputs).

A circle, for instance, fails the vertical line test because a vertical line through the center crosses the curve twice. A parabola opening upward passes the test because each vertical line crosses it at most once.

Graphs of Common Functions

Linear functions: $f(x) = mx + b$

• Graph is a straight line with slope $m$ and y-intercept $(0, b)$.
• The slope $m$ represents a constant rate of change. Between any two points on the line, the ratio $\frac{\Delta y}{\Delta x}$ is the same.

Quadratic functions: $f(x) = ax^2 + bx + c$

• Graph is a parabola. If $a > 0$, it opens upward; if $a < 0$, it opens downward.
• The vertex (the turning point) has its $x$-coordinate at $x = -\frac{b}{2a}$. Plug that back in to find the $y$-coordinate.
• The parabola is symmetric about the vertical line $x = -\frac{b}{2a}$.

Exponential functions: $f(x) = a \cdot b^x$

• If $b > 1$, the function grows rapidly as $x$ increases. If $0 < b < 1$, it decays toward zero.
• The y-intercept is always $(0, a)$ because $b^0 = 1$.
• The graph approaches the x-axis but never touches it (the x-axis is a horizontal asymptote).

Piecewise functions use different formulas on different intervals of the domain. For example:

$f(x) = \begin{cases} x + 1 & \text{if } x < 0 \\ x^2 & \text{if } x \geq 0 \end{cases}$

When graphing any function, look for key features: intercepts, vertices, asymptotes, and intervals where the function is increasing, decreasing, or constant.

Function Properties and Operations

Domain is the set of all valid inputs for a function. For most polynomials, the domain is all real numbers. Watch for restrictions like division by zero or square roots of negatives.

Range is the set of all outputs the function actually produces. For $f(x) = x^2$, the range is $[0, \infty)$ because squaring never gives a negative result.

Codomain is the broader set a function's outputs are allowed to land in. The range is always a subset of the codomain. In College Algebra, you'll mostly focus on domain and range, but knowing the term codomain helps if you see it on an exam.

Composition combines two functions by feeding the output of one into the input of another. It's written $(f \circ g)(x) = f(g(x))$.

To evaluate a composition:

1. Start with the inner function. Compute $g(x)$ first.
2. Take that result and plug it into $f$.

For example, if $f(x) = x^2$ and $g(x) = x + 3$, then:

$(f \circ g)(2) = f(g(2)) = f(5) = 25$

Note that order matters: $(f \circ g)(x)$ and $(g \circ f)(x)$ usually give different results.