Function Concepts & Notation

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Understand what a function is and how it behaves.
  • Interpret and use function notation such as \(f(x)\).
  • Distinguish functions from non-functions using graphs and tables.

Key Ideas

A function is a rule that assigns exactly one output to each valid input.

Key points:

  • Written as \(f(x)\) and read “\(f\) of \(x\).”
  • The input is the independent variable; the output depends on it.
  • A graph represents a function if and only if it passes the vertical line test.

Common Problem Types

1. Evaluating Functions

Plug a value into the expression to find the output.

2. Determining Whether a Relation Is a Function

Check whether each input produces only one output.

3. Using Tables, Graphs, or Mappings

Use the vertical line test for graphs; check for repeated inputs in tables.

4. Reading Function Notation

Expressions like \(f(a+2)\) or \(g(3)\) simply mean “evaluate at that input.”

Strategies

  • Treat \(f(x)\) as a label, not multiplication—it’s just the name of the function.
  • Replace \(x\) with the given input everywhere inside the expression.
  • When checking a graph, imagine sliding a vertical line across it.
  • For mapping diagrams, look for repeated inputs with different outputs.
  • For equations, solve for \(y\) and check if any \(x\) produces multiple \(y\) values.

Worked Examples

Example 1 — Evaluate a Function

Given: \[ f(x) = 3x - 7 \] Find \(f(2)\).

Solution:
\[ \begin{split} f(2) &= 3(2) - 7 \\ &= -1 \end{split} \]

Example 2 — Is It a Function?

Relation shown:

Input → Output
1 → 4
1 → 5

Explanation:
Input 1 has two different outputs, so the relation is not a function.

WarningCommon Mistakes
  • Interpreting \(f(x)\) as multiplication instead of a function label.
  • Mixing up inputs and outputs when evaluating.
  • Believing a relation is a function even when a single input maps to multiple outputs.

Practice Problems

  1. If \(g(x) = x^2 + 1\), find \(g(3)\).
  2. Does \(x^2 = y\) represent a function?
  3. Is a circle a function?
  4. If \(f(t) = 5 - 2t\), find \(f(-3)\).
  5. Is the mapping \(a \to 3\) and \(a \to 4\) a function?

1.
\[ g(3) = 3^2 + 1 = 10 \]

2.
Yes—each \(x\) produces exactly one \(y\).

3.
No—vertical lines hit the circle twice, so it fails the vertical line test.

4.
\[ f(-3) = 5 - 2(-3) = 11 \]

5.
No—a single input (\(a\)) maps to two outputs.

Summary

  • A function assigns exactly one output to each input.
  • Use \(f(x)\) to denote the value of the function at input \(x\).
  • Check functions using the vertical line test for graphs.
  • A relation is not a function if an input maps to multiple outputs.
  • Function notation represents evaluation, not multiplication.
  • \(f(x)\) is a name, not a product—treat it like “the value of the function.”
  • Use vertical line test for graphs; check repeated inputs for tables.
  • Plug in inputs carefully when evaluating expressions.
  • If one input has two outputs, it’s automatically not a function.