Chapter 2: Functions and Graphs

A first pass through function notation, domain, range, graph interpretation, transformations, composition, and inverse functions.

What Is a Function?

Line Slider LabMini lab placeholder for slope and intercept. Forbidden DenominatorDomain restrictions from division by zero.

A function is a rule that assigns each allowed input exactly one output. The input is often called x, and the output is often written as f(x).

input → function rule → output

For example, if f(x) = 2x + 3, then the rule says to double the input and add 3.

Core Concept

A function may use many inputs, but each single input must lead to exactly one output.

Function Notation

The notation f(4) means “the output of f when the input is 4.” It does not mean multiplication by f.

Worked Example 2.2.1 — Substitution Table

Let f(x) = x^2 - 5x + 2. Find f(3).

f(3) = 3^2 - 5(3) + 2
= 9 - 15 + 2
= -4
Core Concept

To evaluate a function, replace every copy of the variable with the given input, then simplify.

Linear Function Mini Lab

A later version of this lab will use sliders for m and b in y=mx+b. The first visual reminds us that changing slope rotates the line while changing intercept shifts it.

Line Slider Lab

y = 1x + 0

Domain and Range

The domain is the set of allowed inputs. The range is the set of possible outputs. In many algebra problems, domain restrictions come from denominators, square roots, and real-world context.

  • Denominators cannot equal zero.
  • Even roots, such as square roots, cannot have negative radicands in the real-number system.
  • Context can restrict inputs, such as time, distance, or number of items.

Worked Example 2.3.1 — Forbidden Denominator

Find the domain of g(x) = 1 / (x - 6).

x - 6 ≠ 0
x ≠ 6

So the domain is all real numbers except 6.

Core Concept

For rational functions, find the input values that make the denominator zero and exclude them from the domain.

Reading Graphs

A graph shows input-output pairs visually. The horizontal axis represents inputs, and the vertical axis represents outputs. To estimate f(2), we move to x = 2 on the horizontal axis and read the graph’s height.

Core Concept

On a function graph, x-values are inputs and y-values are outputs.

The Vertical Line Test

A graph represents a function if no vertical line crosses it more than once. If one input has two different outputs, the graph fails the test.

Basic Transformations

Transformations move or reshape a familiar graph. If f(x) is the parent function, then common changes include vertical shifts, horizontal shifts, reflections, and stretches.

  • f(x) + k moves the graph up k units.
  • f(x - h) moves the graph right h units.
  • -f(x) reflects the graph across the x-axis.
  • f(-x) reflects the graph across the y-axis.
Core Concept

Changes outside the function affect outputs; changes inside the function affect inputs and often feel reversed.

Composition of Functions

Composition means using one function’s output as another function’s input. The notation (f ∘ g)(x) means f(g(x)).

Worked Example 2.6.1 — Inside Then Outside

Let f(x) = 2x + 1 and g(x) = x^2. Find f(g(3)).

g(3) = 3^2 = 9
f(g(3)) = f(9)
f(9) = 2(9) + 1 = 19
Core Concept

Work from the inside out: evaluate the inner function first, then feed that result into the outer function.

Inverse Functions

An inverse function reverses what the original function does. If f(4) = 11, then the inverse sends 11 back to 4.

Worked Example 2.7.1 — Undoing a Linear Rule

Find the inverse of f(x) = 3x - 5.

y = 3x - 5
x = 3y - 5
x + 5 = 3y
y = (x + 5) / 3

So f^{-1}(x) = (x + 5) / 3.

Core Concept

To find an inverse algebraically, swap x and y, then solve the new equation for y.